Introduction to Soliton Theory: Applications to Mechanics Fundamental Theories of Physics
An International Book Series on The Fundamental Theories of Physics: Their Clarification, Development and Application
Editor: ALWYN VAN DER MERWE, University of Denver, U.S.A.
Editorial Advisory Board: GIANCARLO GHIRARDI, University of Trieste, Italy LAWRENCE P. HORWITZ, Tel-Aviv University, Israel BRIAN D. JOSEPHSON, University of Cambridge, U.K. CLIVE KILMISTER, University of London, U.K. PEKKA J. LAHTI, University of Turku, Finland ASHER PERES, Israel Institute of Technology, Israel EDUARD PRUGOVECKI, University of Toronto, Canada FRANCO SELLERI, Università di Bara, Italy TONY SUDBURY, University of York, U.K. HANS-JÜRGEN TREDER, Zentralinstitut für Astrophysik der Akademie der Wissenschaften, Germany
Volume 143 Introduction to Soliton Theory: Applications to Mechanics
by Ligia Munteanu Institute of Solid Mechanics, Romanian Academy, Bucharest, Romania and Stefania Donescu Technical University of Civil Engineering, Department of Mathematics, Bucharest, Romania
KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW eBook ISBN: 1-4020-2577-7 Print ISBN: 1-4020-2576-9
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Preface ix
Part 1. INTRODUCTION TO SOLITON THEORY 1. MATHEMATICAL METHODS 1
1.1 Scope of the chapter 1 1.2 Scattering theory 1 1.3 Inverse scattering theory 12 1.4 Cnoidal method 17 1.5 Hirota method 25 1.6 Linear equivalence method (LEM) 31 1.7 Bäcklund transformation 39 1.8 Painlevé analysis 46
2. SOME PROPERTIES OF NONLINEAR EQUATIONS 53 2.1 Scope of the chapter 53 2.2 General properties of the linear waves 53 2.3 Some properties of nonlinear equations 59 2.4 Symmetry groups of nonlinear equations 62 2.5 Noether theorem 66 2.6 Inverse Lagrange problem 69 2.7 Recursion operators 73
3. SOLITONS AND NONLINEAR EQUATIONS 78 3.1 Scope of the chapter 78 3.2 Korteweg and de Vries equation (KdV) 78 3.3 Derivation of the KdV equation 86 vi INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS
3.4 Scattering problem for the KdV equation 90 3.5 Inverse scattering problem for the KdV equation 95 3.6 Multi-soliton solutions of the KdV equation 101 3.7 Boussinesq, modified KdV and Burgers equations 107 3.8 The sine-Gordon and Schrödinger equations 112 3.9 Tricomi system and the simple pendulum 115 121 Part 2. APPLICATIONS TO MECHANICS 4. STATICS AND DYNAMICS OF THE THIN ELASTIC ROD 121 4.1 Scope of the chapter 121 4.2 Fundamental equations 122 4.3 The equivalence theorem 132 4.4 Exact solutions of the equilibrium equations 134 4.5 Exact solutions of the motion equations 146
5. VIBRATIONS OF THIN ELASTIC RODS 149 5.1 Scope of the chapter 149 5.2 Linear and nonlinear vibrations 149 5.3 Transverse vibrations of the helical rod 155 5.4 A special class of DRIP media 159 5.5 Interaction of waves 163 5.6 Vibrations of a heterogeneous string 166
6. THE COUPLED PENDULUM 173 6.1 Scope of the chapter 173 6.2 Motion equations. Problem E1 173 6.3 Problem E2 177 6.4 LEM solutions of the system E2 180 6.5 Cnoidal solutions 185 6.6 Modal interaction in periodic structures 191
7. DYNAMICS OF THE LEFT VENTRICLE 197 7.1 Scope of the chapter 197 7.2 The mathematical model 198 7.3 Cnoidal solutions 206 7.4 Numerical results 209 7.5 A nonlinear system with essential energy influx 213 CONTENTS vii
8. THE FLOW OF BLOOD IN ARTERIES 220 8.1 Scope of the chapter 220 8.2 A nonlinear model of blood flow in arteries 221 8.3 Two-soliton solutions 228 8.4 A micropolar model of blood flow in arteries 235
9. INTERMODAL INTERACTION OF WAVES 242 9.1 Scope of the chapter 242 9.2 A plate with Cantor-like structure 243 9.3 The eigenvalue problem 248 9.4 Subharmonic waves generation 249 9.5 Internal solitary waves in a stratified fluid 255 9.6 The motion of a micropolar fluid in inclined open channels 259 9.7 Cnoidal solutions 265 9.8 The effect of surface tension on the solitary waves 269
10. ON THE TZITZEICA SURFACES AND SOME RELATED 273 PROBLEMS 10.1 Scope of the chapter 273 10.2 Tzitzeica surfaces 273 10.3 Symmetry group theory applied to Tzitzeica equations 276 10.4 The relation between the forced oscillator and a Tzitzeica 283 curve 10.5 Sound propagation in a nonlinear medium 285 10.6 The pseudospherical reduction of a nonlinear problem 291 References 298 Index 305 This page intentionally left blank Preface
This monograph is planned to provide the application of the soliton theory to solve certain practical problems selected from the fields of solid mechanics, fluid mechanics and biomechanics. The work is based mainly on the authors’ research carried out at their home institutes, and on some specified, significant results existing in the published literature. The methodology to study a given evolution equation is to seek the waves of permanent form, to test whether it possesses any symmetry properties, and whether it is stable and solitonic in nature. Students of physics, applied mathematics, and engineering are usually exposed to various branches of nonlinear mechanics, especially to the soliton theory. The soliton is regarded as an entity, a quasi-particle, which conserves its character and interacts with the surroundings and other solitons as a particle. It is related to a strange phenomenon, which consists in the propagation of certain waves without attenuation in dissipative media. This phenomenon has been known for about 200 years (it was described, for example, by the Joule Verne's novel Les histoires de Jean Marie Cabidoulin, Éd. Hetzel), but its detailed quantitative description became possible only in the last 30 years due to the exceptional development of computers. The discovery of the physical soliton is attributed to John Scott Russell. In 1834, Russell was observing a boat being drawn along a narrow channel by a pair of horses. He followed it on horseback and observed an amazing phenomenon: when the boat suddenly stopped, a bow wave detached from the boat and rolled forward with great velocity, having the shape of a large solitary elevation, with a rounded well-defined heap of water. The solitary wave continued its motion along the channel without change of form or velocity. The scientist followed it on horseback as it propagated at about eight or nine miles an hour, but after one or two miles he lost it. Russell was convinced that he had observed an important phenomenon, and he built an experimental tank in his garden to continue the studies of what he named the wave of translation. The wave of translation was regarded as a curiosity until the 1960s, when scientists began to use computers to study nonlinear wave propagation. The discovery of mathematical solutions started with the analysis of nonlinear partial differential equations, such as the work of Boussinesq and Rayleigh, independently, in the 1870s. Boussinesq and Rayleigh explained theoretically the Russell observation and later reproduction in a laboratory experiment. Korteweg and de Vries derived in 1895 the equation for water waves in shallow channels, and confirmed the existence of solitons. x INTRODUCTION TO SOLITON THEORY : APPLICATIONS TO MECHANICS
An explosion of works occurred when it was discovered that many phenomena in physics, electronics, mechanics and biology might be described by using the theory of solitons. Nonlinear mechanics is often faced with the unexpected appearance of chaos or order. Within this framework the soliton plays the role of order. The discovery of orderly stable pulses as an effect of nonlinearity is surprising. The results obtained in the linear theory of waves, by ignoring the nonlinear parts, are most frequently too far from reality to be useful. The linearisation misses an important phenomenon, solitons, which are waves, which maintain their identity indefinitely just when we most expect that dispersion effects will lead to their disappearance. The soliton as the solution of the completely integrable partial differential equations are stable in collision process even if interaction between the solitons takes place in a nonlinear way. The unexpected results obtained in 1955 by Fermi, Pasta and Ulam in the study of a nonlinear anharmonic oscillator, generate much of the work on solitons. Their attempt to demonstrate that the nonlinear interactions between the normal modes of vibrations lead to the energy of the system being evenly distributed throughout all the modes, as a result of the equipartition of energy, failed. The energy does not spread throughout all the modes but recollect after a time in the initial mode where it was when the experiment was started. In 1965, Zabusky and Kruskal approached the Fermi, Pasta and Ulam problem from the continuum point of view. They rederived the Korteweg and de Vries equation and found its stable wave solutions by numerical computation. They showed that these solutions preserve their shape and velocities after two of them collide, interact and then spread apart again. They named such waves solitons. Gardner, Green, Kruskal and Miura introduced in 1974 the Inverse Scattering Transform to integrate nonlinear evolution equations. The conserved features of solitons become intimately related to the notion of symmetry and to the construction of pseudospherical surfaces. The Gauss–Weingarten system for the pseudospherical surfaces yields sine-Gordon equation, providing a bridge to soliton theory. A privileged surface related to the certain nonlinear equations that admit solitonic solutions, is the Tzitzeica surface (1910). Developments in the geometry of such surface gave a gradual clarification of predictable properties in natural phenomena. A remarkable number of evolution equations (sine-Gordon, Korteweg de Vries, Boussinesq, Schrödinger and others) considered by the end of the 19th century, radically changed the thinking of scientists about the nature of nonlinearity. These equations admit solitonic behavior characterized by an infinite number of conservation laws and an infinite number of exact solutions. In 1973, Wahlquist and Estabrook showed that these equations admit invariance under a Bäcklund transformation, and possess multi-soliton solutions expressed as simple superposition formulae relating explicit solutions among themselves. The theory of soliton stores the information on some famous equations: the Korteweg de Vries equation, the nonlinear Schrödinger equation, the sine-Gordon equation, the Boussinesq equation, and others. This theory provides a fascinating glimpse into studying the nonlinear processes in which the combination of dispersion and nonlinearity together lead to the appearance of solitons.
This book addresses practical and concrete resolution methods of certain nonlinear equations of evolution, such as the motion of the thin elastic rod, vibrations of the initial deformed thin elastic rod, the coupled pendulum oscillations, dynamics of the left PREFACE xi ventricle, transient flow of blood in arteries, the subharmonic waves generation in a piezoelectric plate with Cantor-like structure, and some problems of deformation in inhomogeneous media strongly related to Tzitzeica surfaces. George Tzitzeica is a great Romanian geometer (1873–1939), and the relation of his surfaces to the soliton theory and to certain nonlinear mechanical problems has a long history, owing its origin to geometric investigations carried out in the 19th century. The present monograph is not a simple translation of its predecessor which appeared at the Publishing House of the Romanian Academy in 2002. Major improvements outline the way in which the soliton theory is applied to solve some engineering problems. In each chapter a different problem illustrates the common origin of the physical phenomenon: the existence of solitons in a solitonic medium.
The book requires as preliminaries only the mathematical knowledge acquired by a student in a technical university. It is addressed to both beginner and advanced practitioners interested in using the soliton theory in various topics of the physical, mechanical, earth and life sciences. We also hope it will induce students and engineers to read more difficult papers in this field, many of them given in the references.
Authors This page intentionally left blank PART 1
INTRODUCTION TO SOLITON THEORY
Chapter 1
MATHEMATICAL METHODS
1.1 Scope of the chapter This chapter introduces the fundamental ideas underlying some mathematical methods to study a certain class of nonlinear partial differential equations known as evolution equations, which possess a special type of elementary solution. These solutions known as solitons have the form of localized waves that conserve their properties even after interaction among them, and then act somewhat like particles. These equations have interesting properties: an infinite number of local conserved quantities, an infinite number of exact solutions expressed in terms of the Jacobi elliptic functions (cnoidal solutions) or the hyperbolic functions (solitonic solutions or solitons), and the simple formulae for nonlinear superposition of explicit solutions. Such equations were considered integrable or more accurately, exactly solvable. Given an evolution equation, it is natural to ask whether it is integrable, or it admits the exact solutions or solitons, whether its solutions are stable or not. This question is still open, and efforts are made for collecting the main results concerning the analysis of nonlinear equations. Substantial parts of this chapter are based on the monographs of Dodd et al. (1982), Lamb (1980), Drazin (1983), Drazin and Johnson (1989), Munteanu and Donescu (2002), Toma (1995) and on the articles of Hirota (1980) and Osborne (1995).
1.2 Scattering theory Historically, the scattering theory was fairly well understood by about 1850. It took almost one hundred years before the inverse scattering theory could be applied. Since 1951, various types of nonlinear equations with a soliton as a solution have been solved by direct and inverse scattering theories. However, given any evolution equation, it is natural to ask whether it can be solved in the context of the scattering theory. This question is related to the Painlevé property. We may say that a nonlinear 2 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS partial differential equation is solvable by inverse scattering technique if, and only if, every ordinary differential equation derived from it, by exact reduction, satisfies the Painlevé property (Ablowitz et al.). The Painlevé property refers to the absence of movable critical points for an ordinary differential equation. Let us begin with the equation known as a Schrödinger equation, of frequent occurrence in applied mathematics (Lamb)
MOxx [(,)]uxt M 0 , (1.2.1) where Mo:R R is a dimensionless scalar field in one space coordinate x . The potential function uxt(,) contains a parameter t , that may be the temporal variable, t t 0 . At this point, t is only a parameter, so that the shape of ux(,)t varies from t . Subscripts that involve x or t are used to denote partial derivatives, for example wu wu u , u . t wt x wx If the function u depends only on x , axbd d , where a and b can be infinity, the equation (1.2.1) for imposed boundary conditions at x a and b , leads to certain values of the constant O (the eigenvalues O j ) for which the equation has a nonzero solution (the eigenfunctions M j ()x ). For a given function ux() , the determination of the dependence of the solution M on the parameter O and the dependence of the eigenvalues O j on the boundary conditions is known as a Sturm-Liouville problem. The solutions of (1.2.1) exist only if b the function ux() is integrable, that is |()|dux x f. The spectrum of eigenvalues O ³ j a is made up of two cases corresponding to O!0 and O 0 . The case O 0 does not occur if ux()z 0. In particular, for ux( ) 2sech2 x , and the boundary conditions M()0rf leads to the single eigenvalue O 1 with the associated eigenfunction M sech x . The scattering solutions of (1.2.1) are made up of linear combinations of the functions
M1 exp(i Ox )(iO tanhx ) , and M2 exp( i Ox )(i O tanhx ) . The solving of the Schrödinger equation (1.2.1) when the potential function ux() is specified is referred to as the direct scattering problem. If u depends on x and t , uuxt (,), then we expect the values of the O j to depend upon t . It is interesting to ask whether or not there are potential functions uxt(,) for which the O j remain unchanged as the parameter t is varied.
In particular, if uuxt ( ) satisfies the linear partial differential equation uux t , the variation of t has no effect upon the eigenvalues O j . Also, the eigenvalues are invariant to the variation of t , if ux(,)t satisfies the nonlinear partial differential equation
uuuutxxxx 0 , (1.2.2) MATHEMATICAL METHODS FOR NONLINEAR EQUATIONS ANALYSIS 3 known as the Korteweg–de Vries equation (KdV) . Therefore, solving the KdV equation is related to finding the potentials in a Sturm- Liouville equation, and vice versa. The direct scattering problem is concerned with determining of a wave function M when the potential u is specified. Determination of a potential u from information about the wave function M is referred to as the inverse scattering problem.
THEOREM 1.2.1 LetS be a pre-hilbertian space of functions y :R2 o R . Let us consider the operators LS: o S , BS: o S having the properties:
a) Ly12,, y y 1 Ly 2, yy12, S. b) L admits only simple eigenvalues, namely O()t is an eigenvalue for L if there exists the function <S , so that Lxttx,(,)()(,< O c) Baty,() at () By , , y S , and at() y S. It follows that the relations LLBBLt 0 , (1.2.4) LS: S , Lyxt , (,) Ly , (,) xt Lyxt , (,), (1.2.5) tto ,t t are verified. Also, it follows that 1. the eigenvalues are constants O()t O R, t R , (1.2.6) 2. the eigenfunctions verify the evolution equation Lxttx,(,)()(,< O LxtLxttxttxttt,<< (,) , (,) O<O< () (,) ()t (,)t , LxtLxttxttxttt,<< (,) , (,) O<O< () (,) ()t (,)t , xt,R. From (1.2.4) and (1.2.5) it results O This implies Ot 0 . The function < < Mt B,M . (1.2.8) To illustrate this, let us consider the example ^`yy:R×R+ ooof R, ,yxx 0, , with the scalar product f yy,(,)( yxtyxt,)dx, 12S ³ 1 2 f and operators LB,: So S Ly,( yxx uxty,) , (1.2.9) B,46(,)3(, y yxxx uxt yx u x xt) y. (1.2.10) According to Lytt,(, uxty) , 2 L, By 4 yxxxxx 10 uy xxx 15 u x y xx 12 u xx y x 6 u yx 3 u xxx y 3 uux y , 2 B, Ly 4 yxxxxx 10 uy xxx 15 u x y xx 12 u xx y x 6 u yx 4 u xxx y 9 uux y , it is found that (1.2.3) can be written under the standard form of the KdV equation uutxxxx60 uu . The operators satisfy the properties mentioned in the theorem 1.2.1. For (1.2.5) we find <O Mxx O()u M 0 , (1.2.11) which verifies the equation Mtx 463M xxx uuM xM . (1.2.12) Consequently, finding solutions to the KdV equation is related to solving the Schrödinger equation 2 Mxx [(,)]kuxtM 0 . (1.2.13) Note that in (1.2.13), t is playing the role of a parameter, k is a real or a pure complex number ik ,k ! 0 , and the potential function u has the property uxoof0, . For localized potentials ux() , all solutions of (1.2.13) will reduce to a linear combination of the functions exp(ikx ) , and exp( ikx ) as x of. Following Faddeev (1967), the solutions of the Schrödinger equation are expressed as linear combinations of a solution fxk1 (,) that reduces to exp(ikx ) , as x of, and a solution fxk2 (, ) that reduces to exp(i kx ), as x of. By definition, fxk1 (,) and fxk2 (, ) are fundamental solutions of (1.2.13) and are exact solutions of (1.2.13) and verify fxk1 (,)|o exp(i), kxx f, fxk1 (,)exp(io kx ) 1, x of, (1.2.14a) fxk2 (, )| exp(i kxx ), of, fxk2 (,)exp(i) kxoof 1, x . (1.2.14b) THEOREM 1.2.2 Fundamental solutions fxk1 (,) andfxk2 (, ) verify the equations 1 f fxk(, ) exp(i) kx sin( kx DDD )( u ) f (, k )dD , (1.2.15a) 11³ k x 1 x fxk(,) exp(i kx ) sin( kx DDD )( u ) f (,)d kD . (1.2.15b) 21³ k f Proof. The homogeneous equation associated to (1.2.13), M'' k 2M 0 , admits the solutions M()x A ()exp(i) x kx B ()exp(i x kx ), with A, B arbitrary constants. By applying the method of variation of constants, we obtain 1 A'(xuxx ) ( )M ( ) exp( ikx ) , 2ik 6 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 1 Bx'( ) ux ( )M ( x ) exp(i kx ) , 2ik and then, by integration, we have 1 x A()xu ()()exp(iDM DDD k )d C , ³ 1 2ik 0 1 x Bx( ) u ( D )M (DDD )exp(ik )d C . ³ 2 2ik 0 The constants CC1, 2 are found from (1.2.14a) 1 f Cu 1()()exp(i DM DD k)dD , 1 ³ 2ik 0 1 f Cu ()()exp(i)dDM DD kD . 2 ³ 2ik 0 Substitution of these expressions into M()x , leads to 1 f M(xk , ) exp(i kx )D³ u ( )M (DD )expi kx ( )dD 2ik x 1 f DMDDD³ukx()()expi( )d. 2ik x The function f2 is derived in an analogous manner. Equations (1.2.15) are the Volterra integral equations, which can be solved by an iteration procedure. More specifically, the substitution of exp(ikx ) into (1.2.15a) yields to the conclusion that the resulting integrals converge for Im(k )! 0 . For integral equations of Volterra, the resulting series expansion is always convergent. Hence, the functions ff12, are analytic in the upper half of the complex k plane. For real ux() and k , we have fii(,xk ) fxk (, ),i 1, 2 , where “ ” is the complex conjugate operator. From (1.2.15) we see that the functions fxk1 (,),fx1 (, k ) are independent. The functions fxk2 (, ),fxk2 (, )are also independent. So, there exist the coefficients ckij (), ij,1,2 , depending on k, so that fxk21(,) ckfxkckfxk111 () (,)21 () (, ), (1.2.16a) fxk12(,) c122 () kfxk (,) c22 () kfx (, k ). (1.2.16b) From the limiting form of fxi (,r k ), i 1,2, we may write fxk2 (,)| exp(i kxx ),of , MATHEMATICAL METHODS FOR NONLINEAR EQUATIONS ANALYSIS 7 fxk21( , )| ck1 ( )exp(i kxck ) 12 ( )exp(o i kxx ),f , which means the solution corresponds to a scattering problem in which the incident wave is coming from f with an amplitude ck12 () , and is reflected with an amplitude ck11 (), and transmitted to f with an amplitude of unity. In particular, the fundamental solutions for the potential ux( ) 2sech2 x, are obtained by solving the equation zkcc (2sech)22 xz , f x f. By using the substitution yx tanh , 11x , we obtain the associated Legendre equation (Drazin and Johnson) ddzk2 [(1yz2 ) ] ( 2 ) 0 , dd1yy y2 whose general solution is given by zA exp( kxkyB )( ) exp( kxky )( ) . From here we obtain the fundamental solutions 1 fxk( , ) exp(i kxk )(i tanh x ) , 1 i1k 1 fxk( , ) exp( ikxk )(i tanh x ) . 2 i1k Let us introduce the reflection and transmission coefficients for an incident wave of unit amplitude (Achenbach). The ratio ck11 () RkR () , (1.2.17) ck12 () is the reflection coefficient at f , and the ratio 1 TkR () , (1.2.18) ck12 () is the transmission coefficient at f . The subscript R refers to a wave incident from the right. Similarly, we have fxk1 (,)|o exp(i), kxx f , fxk12( , )| c1 ( k )exp(i kxc ) 22 ( k )exp(o i kx ), x f , which means the incident wave from f with an amplitude ck21 () is reflected with an amplitude c22 ()k and transmitted to f with an amplitude of unity. The ratio 8 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS ck22 () RkL () , (1.2.19) ck21 () is the reflection coefficient at f , and the ratio 1 TkL () , (1.2.20) ck21 () is the transmission coefficient al f . The Wronskian of any two functions M1 and M2 , is defined as wx[(),()]MM12 x MM 1,2x ()() xx MM 12, () xx () x. (1.2.21) If M1 and M2 are two linearly independent solutions of (1.2.13), then their Wronskian is a constant wxkxkfk[MM12 (;),(;)] (), k C . (1.2.22) The relation (1.2.22) results by adding (1.2.14) written for M1 and multiplied by M2 , to (1.2.13) written for M2 and multiplied by M1 . It results d w[,MM ]0 . dx 12 According to definition of the Wronskian and (1.2.22), the following properties hold wf[(;),(;)]2i11 xk f x k k, wf[(;),(;)]2i22 xk f x k k, (1.2.23) whereff12, are fundamental solutions (1.2.15). Substituting fxk1 (; ) from (1.2.16b) into (1.2.16a), and substituting fxk2 (; ) from (1.2.16a) into (1.2.16b), k C , and taking account of the independency of fxkfx22(; ),(,k) , the following relations are obtained ckck11() 22 () ckc 12 () 21 ( k ) 1, ckck11() 21 () ckc 12 () 22 ( k ) 0, (1.2.24) ckck21() 12 ( ) ckck22 () 11 () 1, ckck21() 11 ( ) ckck22 () 12 () 0. The coefficients cij may be written in terms of the Wronskian 1 ck( ) wfxkfxk [ ( ; ), ( ; )], 11 2ik 2 1 1 ck() wfxkfxk [ (; ),(;)], (1.2.25) 22 2ik 2 1 MATHEMATICAL METHODS FOR NONLINEAR EQUATIONS ANALYSIS 9 1 ck( ) ck ( ) wfxkfxk [ ( ; ), ( ; )], 12 21 2ik 1 2 k C . That yields TkRL() Tk () Tk (). The relations (1.2.25) are obtained from (1.2.16) and (1.2.23). Furthermore, from fxkii(; ) fxki (; ), 1,2, we have RkTkRL()( ) R ( kTk )() 0, k C , (1.2.26) and ckck12 () 12 (), ck11 () c22 () k c22 ( k ), 22 2 ck12 () 1 ck11 () 1 ck22 () , (1.2.27) 2222 Tk() RRL () k Tk () R () k 1, RkRkRkRkLLRR() (),() (), k R . The location of the poles of the transmission and reflection coefficients in the upper half-plane are important to obtain information about the localized or bound-state solutions. Consider now the poles of Tk() . THEOREM 1.2.3 For real potential functions u :Ro R , any poles of the transmission coefficient in the upper half-plane must be on the imaginary axis. More precisely, if kC0 is a pole for Tk(), then k00 i,N N0 R . 1 Proof. Let k0 C be a pole for Tk() . Then it is a zero for c12 , ck12 () ck12() 0 0. According to (1.2.27) 3 , for k R we have ck12 ()z 0, and then Imk0 ! 0 . Writing (1.2.13) for k0 2 M''M (ku0 ) 0 , and similarly for k0 2 M ''M (ku0 ) 0 , and subtracting them, it follows that 22 MMMM'' '' (kk00 ) MM. Integrating then over x , from f to f , yields f 22 2 ww(,MM ) (, MM ) ( kk00 ) M dx . f f ³ f The Wronskian of MM, being a constant, it follows that 10 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS f 2 Rekk ImM d x 0 , 00³ f and hence, from Im k0 ! 0 , it results Rek0 0 . When ck12() 0 0 ,kC0 , the fundamental solutions are linearly dependent, and then lead to fxk2011010(, ) ckfxk ( ) (, ), (1.2.28) and 1 ck22() 0 . (1.2.29) ck11() 0 This property results from (1.2.16a,b) written for k k0 . Next, we show that the value of the residuum of the function Tk() in every pole klll i,N N! 0 is given by 1i Res(Tk ( ))( k ) . (1.2.30) l ck () f 12 l fxkfxkx(, ) (, )d ³ 12ll f To obtain this, let us differentiate with respect to k , the relation (1.2.25)3 and set kk l . According to (1.2.28) and (1.2.29) we have d1 1 ck() ckwxk ( ) (, ) ckwxk ( ) (, ), (1.2.31) d2i2ikk12 11ll 1 k22 l 2 l kk l ll where wf wxk(, ) w [i (, xk ) ; fxk (, )], i 1,2. ilwk il kk l To obtain w1 , let us multiply (1.2.13) written for fxk1 (,) withfxk1 (,l ) , then multiply (1.2.13) written for fxk1 (,l ) with fxk1 (,) , and add the results. We have ww w [f (, xk ) f (, xk ) f (, xk ) f (, xk )]( k22 k ) f (, xk ) f (, xk ) 0. wwxx1111ll w x l11 l Differentiating the above relation with respect to k R , we have ww wfxk[(,); fxk (,) ]2((,)) kfxk 2 . xk11ll 1l wwkk l Integration from x to f , gives w f Awfxk[(,); fxk (,)]2 k ((,))d fDD k 2 , 11llll³ 1 wk x MATHEMATICAL METHODS FOR NONLINEAR EQUATIONS ANALYSIS 11 w Awfxkfxk lim[11 (,ll ); (, )]0 . xof wk It follows that f wkfk 2[(;)]dDD2 . 11ll³ x In a similar way we obtain x wkfk 2[(;)]dDD2 . 22ll³ f Substitution ofwii ,1, 2 , into (1.2.31) yields f ck11 ()l 2 ck12() [f 1 (;DD kl )]d kk l ³ i f 1 ffck() [(;)](;)dfkfkDDD 22 l [f (;)]d.D k2 D ³³12ll 21l iif f Note that f ck()[(;)]dJ fxk2 x 1, (1.2.32a) ³ 11ll 1 l f f ck()[J fxk (;)]d2 x 1. (1.2.32b) ³ 22ll 2 l f 1/2 1/2 Thus, the quantities [(Jllck11 )] and [(Jllck22 )] are the normalization constants for the bound-state wave functions fxil(,k ),i 1,2 . Using (1.2.30) we may write the normalization constants as ck() f mck J() i11 l {[(;)]d}fxkx2 1 , (1.2.33a) Rl l11 l³ 1 l ck12 ()l f ck() f mck J() i22 l {[fxkx (;)]d}2 1 , (1.2.33b) Ll l22 l³ 2 l ck12 ()l f wheremmRl, Ll R(,)R,i,1,2 , due to the fact that fxkil k l N l i . Any poles of the transmission coefficient are simple because, if kl is a pole for Tk(), kll iN,N!l 0 , then it has the properties (1.2.32a), (1.2.33a), and it results f ck () i ck ()[(;)]d fxk2 xz 0. 12 lll11 ³ 1 f 12 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 1.3 Inverse scattering theory The inverse scattering theory was firstly considered to solve an inverse physical problem of finding the shape of a mechanical object, which vibrates, from the knowledge of the energy or amplitude at each frequencies (Drazin and Johnson). In our terms, the methods consist in determination of the potential function u from given coefficients ckij () , that relate the fundamental solutions of the equation M''M (ku2 ) 0 . (1.3.1) The fundamental solutions of the Schrödinger equation may be written under the form f fxk( , ) exp(i kx ) Axx ( , ')exp(i kxx ')d ' , (1.3.2a) 1 ³ R x x fxk(,) exp(i) kx Axx (, ')exp(i')d kxx' . (1.3.2b) 2 ³ L f Balanis considered these forms in 1972, by solving the elastically braced vibrating string equation yyxx x'' x uxy() 0, (1.3.3) for which the solutions are written as yxx1 (, ') G (' x x ) T (' x xAxx )R (, '), (1.3.4a) yxx2 ( , ') G ( x ' x ) T ( x ' xAx )L ( , x '), (1.3.4b) whereGT is the Dirac function, is the Heaviside function, and AR , AL are functions that describe the scattering or wake. Applying the Fourier transform f F[(,')](,)yxx xk ³ yxx (,')exp(i')d' kx x, f to (1.3.3), we find (Lamb) 2 Fyxx[ ( , ')]xx ( k ux ( )) Fyxx [ ( , ')] 0 . (1.3.5) The equation (1.3.5) admits as solutions the Fourier transform of (1.3.4) f F[yxx ( , ')]( xk , ) exp(i kx ) Axx ( , ')exp(i kxx ')d ' , (1.3.6a) 1 ³ R x x F[yxx ( , ')]( xk , ) exp( i kx ) Ax ( , x ')exp( i kxx ')d ' . (1.3.6b) 2 ³ L f From (1.3.5), we see that the solutions of (1.3.1) take the form (1.3.6) fxkii(, ) Fyxx [ (, ')](, xk ), i 1,2. MATHEMATICAL METHODS FOR NONLINEAR EQUATIONS ANALYSIS 13 Substituting ff12, into (1.3.1) we derive the conditions to be verified by AR , AL . For this, we writef1 as i1wA fxk(,) exp(i)[1 kx Axx (,') R (,')] xx 1 kxRxk 2 w ' ' x 1 f w2 A R (,xx ')exp(i')d', kx x 22³ kxx w ' then integrate it by parts and introduce into (1.3.1). By imposing the conditions AxxRR(, '), Ax' (, xx ')oo 0,for x ' f , we find dA exp(ikx )[2R ( x , x ) u ( x )] dx f ww22AA [RR (,')xx (,')() xx uxA (,')]exp(i')d' xx kx x 0. ³ 22 R x wwxx' Therefore, the equation (1.3.1) is verified for dA ux() 2R (, xxc ), dx AR (,xx ')0, x ' x , (1.3.7a) ww22AA RR(,x xxxuxAxxx ') (, ') () (, ')0, '! x . wwxx22' R Similarly, f2 verifies the equation (1.3.1) for dA ux() 2L (, xxc ), dx AL (,xx ')0, x '! x , (1.3.7b) ww22AA LL(,x xxxuxAxxx ') (, ') () (, ')0, ' x , wwxx22' L and the Faddeev condition is verified (Faddeev 1958) f ³ (1 fxux )()d x . f From (1.3.7) we see that for given AR , AL , we can find the potential function u. Next, we try to determine the functions AR , AL in terms of the coefficients, ckij (),, ij 1,2, considered specified. For this we write (1.2.14a) under the form 14 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS Tk() f211 (, xk ) RR () k f (, xk ) f (, x k ), (1.3.8) and derive the corresponding relation in the time domain. Taking the Fourier transform does this 1 f F 1[(,)](,')fxk xx fxk (,)exp(i kxk ')d. ³ 1 2S f By noting 1 f *()zTkkz ³ [ () 1]exp(i )dk , (1.3.9) 2S f the Fourier transform on the left-hand side of (1.3.8) yields 1 ff F 1[()Tk f (,)](,') xk xx (1* ()exp(i)d)z kzz u 2 ³³ 2S f f f u(yxx ( , '') exp(i kxx '')d '') exp( i kxk ')d ³ 2 f 1 yxx( , ')* ( zyxx ) ( , '') exp[i kzx ( '' x ')]d kxz d ''d 22³³³ 2S f, f G('xx ) T (' xxAxx )L (, ') f (1.3.10) *(x xxxAxx ') * ( ' '') ( , '')dx ''. ³ L x In a similar way, by noting 1 f rz() Rk ()exp(i)d kzk, (1.3.11) RR³ 2S f the Fourier transform on the right-hand side of (1.3.8) leads to 1 FRkfxkfxkxx[R ()11 (,) (, )](,') 1 f Rkfxk() (,)exp(i kxk ')d yxx (, ') ³ R 11 2S f 1 ff f (r ( z )exp(i kz )d z )( y exp(i kx '')d x '')exp( i kx ')d k y ( x , x ') ³³R ³1 1 2S f f f f 1 rzyxx( ) ( , ''){ exp[i k ( z x '' x ')]d k }d x ''d zyx ( , x ') ³³R 11 ³ f, f f 2S f rx(''')(,'')d''(, xyxx x yx x ') ³ R 11 f MATHEMATICAL METHODS FOR NONLINEAR EQUATIONS ANALYSIS 15 f rx( x ') rx ( '' xAxxx ') ( , '')d '' RRR³ x G('x xxxAxx ) T ( ' )R (, '). (1.3.12) Substituting (1.3.10) and (1.3.12) into (1.3.8) we have f T(')(,')(xxAxx * xx ') * (''')(,'')d'' xxAxx x LL³ x (1.3.13) f rx( x ') rx ( '' xAxxx ') ( , '')d '' T ( x ' xAx ) ( , x '). RRR³ R x We study now the case xx '0. To interpret (1.3.13) we must evaluate the function *()z , z 0 . Case 1. When the transmission coefficient Tk() possesses neither poles nor zeros in the upper half-plane, then *()z 0. (1.3.14) To show this, we consider the closed contour in the complex plane CC R [, RR] where CR a semicircle of radius R . According to the Cauchy theorem we have ³[Tk ( ) 1]exp( i kzk )d 0 , C due to the fact that the integrant is an olomorphic function in the simple convex domain enclosed by the contour C . Then, we can write R ³³[Tk ( ) 1]exp( i kzk )d [ Tk ( ) 1]exp( i kzk )d . (1.3.15) CRR According to the Jordan lemma, if CR is a semi-circle in the upper half-plane, centered in zero and having the radius R, and the function Gk() satisfies the condition Gk()oof 0, k , in the upper half-plane and on the real axis, and m is a positive real number, then we have ³ Gk()exp(i)d kmkoo 0, R f . CR Here mz R , Gk() Tk () 1. If Tk()oof 1, k , we may write ³ [Tk ( )oo 1]exp( i kzk )d 0, R f . CR From (1.3.15) it results 16 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS R *()zTkkzk lim[ () 1]exp(i )d 0. Rof ³ R When Tk() contains no poles in the upper half-plane, the equation (1.3.13) becomes f rxx( ') rx ( '' xAxxx ') ( , '')d '' Ax ( , x ') 0, x x ' 0 , RRRR³ x or, denoting x ' y f rx( y ) rx ( '' yAxxx ) ( , '')d '' Axy ( , ) 0, x y. (1.3.16a) RRRR³ x In a similar way we obtain x rx( y ) rx ( '' yAxx ) ( , '')d x '' Axy ( , ) 0, x ! y, (1.3.16b) LLLL³ f where 1 f rz() Rk ()exp(i kzk )d . LL³ 2S f Case 2. If Tk()contains first-order zeros or poles in the upper half-plane, then they are situated on the imaginary axis kllll iN , N! 0, 1,2,..., n . From the residuum theorem we find f 1 n exp( ikz ) *()zTkkzk [ () 1]exp(i )d 2iS l Rez[Tk ( )] ³ ¦ l f 2S l 1 2S nn i¦¦ exp(kzll )iJ exp(kzll )J , (1.3.17) ll 11 whereJl is given by (1.2.31). The relation (1.3.13) is written as nnf JJexp[kx ( x ')] exp[kx ( x '')] Ax ( , x '')d x '' ¦¦ll³ l lL ll11 x (1.3.18) f rxx( ') rx ( '' xAxxx ') ( , '')d '' Ax ( , x '), RRRR³ x forxx '0 . In terms of x ' y , and taking into consideration that MATHEMATICAL METHODS FOR NONLINEAR EQUATIONS ANALYSIS 17 f fx( ,iN ) c (i N ) fx ( ,i N ) c (i N )[exp( Nx ) Axx ( , ')exp(N x '')d x ''], 21lllllR1111 ³ l x the equation (1.3.18) becomes f ::(x y ) ( x '' yAxxx ) ( , '')d '' Axy ( , ) 0 , x y . (1.3.19a) RRRR³ x Similarly, we obtain x ::(x yxyAxxxAxyx ) ( '' ) ( , '')d '' ( , ) 0, !y . (1.3.19b) LLRL³ f The functions :R ()z and :L ()z are defined as n (zrz ) ( ) c (i )exp( kz ) :RR JN¦ ll11 l l 1 (1.3.20a) 1 f ck() n 11 exp(ikz )d kN i m (i )exp(N z ), ³ ¦ Rl l l 2()S f ck12 l 1 1 f ck() n :()zk 22 exp( iz )dk im (i N )exp( Nz ) , (1.3.20b) LL³ ¦ ll l 2()S f ck12 l 1 with mRl and mLl given by (1.2.31) c11 (iNl ) c22 (iNl ) mRl(iN l ) i , mLl(iN l ) i . c12 (iNl ) c12 (iNl ) In this case we have obtained the same integral equations (1.3.19) as in the first case, with the difference that rR is replaced to :R . These equations are known as Marchenko equations (Agranovich and Marchenko), and they can be used to determine AR or AL when one of the reflection coefficients rR or rL is specified. Solutions of Marchenko equations are the functions AR , AL , which allow the determination of the potential function u . The Marchenko equations are also used to determine the reflection coefficients when the potential and hence the fundamental solutions and the functions AR or AL are specified. We can say that determination of RRR , L and rrR , L , is made from (1.2.15) and (1.2.17), and determination of ::R , L from (1.3.20). 1.4 Cnoidal method The inverse scattering theory generally solves certain nonlinear differential equations, which have cnoidal solutions. The mathematical and physical structure of the inverse scattering transform solutions has been extensively studied in both one and two 18 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS dimensions (Osborne, Drazin and Johnson, Ablowitz and Segur, Ablowitz and Clarkson). The theta-function representation of the solutions is describable as a linear superposition of Jacobi elliptic functions (cnoidal functions) and additional terms, which include nonlinear interactions among them. Osborne is suggesting that the method is reducible to a generalization of the Fourier series with the cnoidal functions as the fundamental basis function. This is because the cnoidal functions are much richer than the trigonometric or hyperbolic functions, that is, the modulus m of the cnoidal function, 0 ddm 1 , can be varied to obtain a sine or cosine function (m # 0)), a Stokes function (m # 0.5 or a solitonic function, sech or tanh (m # 1) (Nettel). Since the original paper by Korteweg and DeVries, it remains an open question (Ablowitz and Segur): “if the KdV linearised equation can be solved by an ordinary Fourier series as a linear superposition of sine waves, can the KdV equation itself be solved by a generalization of Fourier series which uses the cnoidal wave as the fundamental basis function?” This method requires brief information necessary to describe the cnoidal waves. The arc length of the ellipse is related to the integral z (1 kx22 )d x Ez() , ³ 2 0 (1 x ) with 0 k 1 . Another elliptical integral is given by z dx Fz() . ³ 222 0 (1x )(1kx ) The integralsEz() and F()z are Jacobi elliptic integrals of the first and the second kinds. Legendre is the first who works with these integrals, being followed by Abel (1802–1829) and Jacobi (1804–1851). Jacobi inspired by Gauss, discovered in 1820 that the inverse of F()z is an elliptical double-periodic integral F 1 ()Z sn(). Z Jacobi compares the integral M dT v , (1.4.1) ³ 21/2 0 (1Tm sin ) where 0 ddm 1 , to the elementary integral M dt w , (1.4.2) ³ 21/2 0 (1 t ) and observed that (1.4.2) defines the inverse of the trigonometric function sin if we use the notations t sin T and \ sin w . He defines a new pair of inverse functions from (1.4.1) snv sin M, cnv cos M. (1.4.3) MATHEMATICAL METHODS FOR NONLINEAR EQUATIONS ANALYSIS 19 These are two of the Jacobi elliptic functions, usually written sn(vm , ) and cn(vm , ) to denote the dependence on the parameter m . The angle M is called the amplitude M amu . We also define the Jacobi elliptic function dnvm (1Msin21 ) / 2 . Form 0 , we have v M, cn(vv ,0) cosM cos , v M sn(vv ,0) sinM sin , dn(v ,0) 1 , (1.4.4) and for m 1 v arcsech(cosM ) , cn(vv ,1) sech , sn(vv ,1) tanh , dn(vv ,1) sech . (1.4.5) The functions sn v and cn v are periodic functions with the period 2/SSddTT2 4. ³³2 1/2 2 1/2 00(1Tmm sin ) (1 T sin ) The later integral is the complete elliptic integral of the first kind S /2 dT Km() . (1.4.6) ³ 21/2 0 (1Tm sin ) The period of the function dn v is 2K . For m 0 we have K(0) S / 2 . For increasing of m , K()m increases monotonically 116 Km()| log . 21 m Thus, this periodicity of sn (v ,1) and cn (vv ,1) sech is lost for m 1 , so Km()of. Some important algebraic and differential relations between the cnoidal functions are given below d cn22 +sn 1, dn22 +m sn 1, cn= sn dn , dv d d sn= cn dn , dn= m sn cn , (1.4.7) dv dv where the argument vm and parameter are the same throughout relations. Now, consider the function ()t introduced by Weierstrass (1815–1897) in 1850, which verifies the equation 23 4g2 g3 , (1.4.8) where the superimposed point means differentiation with respect to t . 20 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 3 Ifeee123,, are real roots of the equation 40ygyg 23 with eee12!!3 , then (1.4.8) can be written under the form 2 4( eee12 )( )( 3 ) , (1.4.9) with 222 g212 2(eee3 ) , g312 4eee3 , ee123e 0 . Introducing 3 2 ' g2 27g3 , (1.4.10) when'!0 , equation (1.4.9) admits the elliptic Weierstrass function as a particular solution, which is reducing in this case to the Jacobi elliptic function cn 2 (;,)(tggeeeGcc23 2G 2 3 )cn( eet 1 3 ) , (1.4.11) whereGc is an arbitrary real constant. If we impose initial conditions to (1.4.9) (0) T0 , c (0) Tp0 , (1.4.12) then a linear superposition of cnoidal functions (1.4.11) is also a solution for (1.4.8) n 2 Tlin 2cn[;¦ Dk Z ktm k ] , (1.4.13) k 0 where the angular frequencies Zk , and amplitudes Dk depend on T0 , T p0 . When ' 0 the solution of (1.4.9) is 1cn(2GtH2 c ) eH22 , 1cn(2GtH2 c ) with 1 3e2 2 g2 m , He22 3 . 24H 2 4 When ' 0 , we have ee12 cc, e3 2 , and the solution of (1.4.9) is 3c c . sinh2 ( 3ct Gc ) Since the calculation of the elliptic functions is very important for practical problems, in Chapter 10, the Shen-Ling method to construct a Weierstrass elliptic function from the solutions of the Van der Pol’s equation is presented. Consider now a generalized Weierstrass equation with a polynomial of n degree in T()t MATHEMATICAL METHODS FOR NONLINEAR EQUATIONS ANALYSIS 21 2 T Pn () T. (1.4.14) The functional form of solutions of (1.4.14) is determined by the zeros of the right- hand side polynomial. For biquadratic polynomial, n 4 , we can have four real zeros, two real and two purely imaginary zeros, four purely imaginary zeros or four genuinely complex zeros. For n 5 the functional form of solutions depends also on the zeros of the polynomial. For all cases the solutions are expressed in terms of Jacobi elliptic functions, the hyperbolic and trigonometric functions. In the following we present the cnoidal method. Osborne discussed this method for integrable nonlinear equations that have periodic boundary conditions, in particular for the KdV equation. In 2002, Munteanu and Donescu have extended this method to nonlinear partial differential equations that can be reduced to Weierstrass equations of the type (1.4.14). We present the method in context of the KdV equation Ttxc0 TDTTET xxxx 0 , (1.4.15) wherec0 ,D and E are constants. ForD 0 , the linearized KdV equation is solved by the Fourier series. The solutions are expressed as a sum of sine waves using the linear dispersion relation 3 Z ck0 Ek . The general solution to the KdV equation with periodic boundary conditions may be written in the terms of the theta function representation (Dubrovin et al.) 2d2 T(xt , ) log 4KKK ( , ,..., ) , (1.4.16) O dx 2 n 12 n whereO D/6 E , and 4 is the theta function defined as nn1 M MB M , (1.4.17) 4KKnn(12 , ,..., K ) ¦¦ exp(i iiiiK ¦jj) Miiff(,) 1 2 ,1j with n the number of degrees of freedom for a particular solution of the KdV equation, and Kj kxjj ZI t j , 1 d jNd . (1.4.18) In (1.4.18), k j are the wave numbers, the Z j are the frequencies and the I j are the phases. Let us introduce the vectors of wave numbers, frequencies and constant phases kkkk [,12 ,...,],n Z [, Z12 Z ,...,], Zn I [ I12 , I ..., In ] , K [, K12 K ,...,] Kn . (1.4.19) The vector K can be written as K kx Z t I. (1.4.20) Also, we can write 22 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS MKxtK : ), M [MM12 , ,..., Mn ] , K Mk , : M Z, ) M I. The integer components in M are the integer indices in (1.4.17). The matrix B can be decomposed in a diagonal matrixDO and an off-diagonal matrix , that is BDO . (1.4.21) THEOREM 1.4.1 (Osborne) The solution T(,)x t of KdV equation (1.4.15) can be written as 2 w2 T(,)xt log 4K () TKTK () (), (1.4.22) O wx2 n lin int whereTlin represents a linear superposition of cnoidal waves 2 w2 TK() log()G K, (1.4.23) lin O wx2 1 GMM(K ) ¦ exp(i K T DM ) , (1.4.24) M 2 andTint represents a nonlinear interaction among the cnoidal waves w2 F (,K C ) TK( ) 2 log(1 ) , (1.4.25) int wKtG2 () 1 F (,KCCMMDM ) ¦ exp(i K T ), (1.4.26) M D 2 1 CMOM exp(T ) 1 . (1.4.27) 2 Proof. The decomposition (1.4.22) result easily from (1.4.16), (1.4.17) and (1.4.21). Consider the case with no interactions ( O 0 , and then C 0 ). The function G()K yields n GGM() ( ), (1.4.28) K mmmK m 1 where f 1 GM() exp(i MD2 ). (1.4.29) mmK ¦ mm K mmm M m f 2 So, the linear term (1.4.23) becomes MATHEMATICAL METHODS FOR NONLINEAR EQUATIONS ANALYSIS 23 n 2 w2 TK() logGM (K ) lin ¦ Owx2 m m m m 1 (1.4.30) n 2 cn2 {(Km ( ) / )[ x Ctm , ]}, ¦ Kmm S mm m 1 O where K()m is the elliptic integral (1.4.6). The relation (1.4.30) provides the interpretation of the first term on the right-hand side of (1.4.22) as a linear superposition of cnoidal waves. The solution (1.4.30) represents the cnoidal wave solution for the KdV equation. The moduli mm and the phase speeds Cm , mn 1,2,..., , are given by 22 mKmm()4 m S Un , 2(K Km2 ) Cc (1mm 2 kh22 ) , mm0 h 3S2 3Km where U m 33is the Ursell number. 8khm Consider now a nonlinear system of equations that govern the motion of a dynamical system dT i Fin(TTT , ,..., ), 1,..., ,n t 3 , (1.4.31) dt in12 with x R n , tT[0, ] , T R , where F may be of the form nn n Fa b c i ¦¦ ippTTTTTT ipqpq ¦ ipqrpqr ppqpqr 1,1, ,1 (1.4.32) nn de TTTTTTTTT¦¦ipqrl p q r l ipqrlm p q r l m ... , pqrl,,, 1 pqrlm,,,, 1 with in 1,2,..., , and abc, , ...constants. The system of equations has the remarkable property that it can be reduced to Weierstrass equations of the type (1.4.14). In the following, we present the cnoidal method, suitable to be used for equations of the form (1.4.31). To simplify the presentation, let us omit the index i and note the solution by T()t . We introduce the function transformation d2 T 2log( 4 t) , (1.4.33) dt 2 n where the theta function 4n ()t are defined as 411 1exp(i ZtB 11 ), 421 1 exp(i ZtB112 ) exp(i Z tB 221 ) exp( ZZ2B12 ) , 24 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 431 1 exp(i ZtB112 ) exp(i Z tB 22 ) ZZZexp(itB ) exp( B ) 3331212 (1.4.34) ZZZZexp(1 3BB 13 ) exp(2 3 23 ) exp( Z123121323 Z Z BBB ), and nn1 4nii ¦¦exp(i M ZtBMM ¦ij i j ) , (1.4.35) Miff(,) 1 2 ij 2 §·ZZ exp B ij, exp B Z2 . (1.4.36) ij ¨¸ ii i ©¹ZZij Further, consider (1.4.21) and write the solution (1.4.33) under the form w2 T()t 2 log 4K () TKTK () (), (1.4.37) wt 2 n lin int for K Zt I. The first term Tlin represents, as above, a linear superposition of cnoidal waves. Indeed, after a little manipulation and algebraic calculus, (1.4.23) gives n 2S f qtk 1/2 SZ T D[[cos(21)l k l ]]2 . (1.4.38) lin ¦¦l 21k lk 10Kmll 1 ql 2Kl In (1.4.38) we recognize the expression (Abramowitz and Stegun, Magnus et al.) n 2 Tlin ¦ D lcn [ Z ltm ; l ] , (1.4.39) l 1 with Kc q exp(S ) , K S /2 du KKm () , ³ 2 0 1-mu sin Kmc()11 Km (), m m 1. The second term Tint represents a nonlinear superposition or interaction among cnoidal waves. We write this term as 2 2 d()Ft EZk cn (tm ,k ) 2log[1]2 | 2 . (1.4.40) d1ttGt() Jkkcn( Z,)m MATHEMATICAL METHODS FOR NONLINEAR EQUATIONS ANALYSIS 25 Ifmk take the values 0 or 1 , the relation (1.4.40) is directly verified. For 7 0 d mk d 1 , the relation is numerically verified with an error of ||e du510 . Consequently, we have n cn2 [tm ; ] ¦EZkkk k 0 Tint (,)xt n . (1.4.41) 1cn[;2 tm] O¦ kk Z k k 0 As a result, the cnoidal method yields to solutions consisting of a linear superposition and a nonlinear superposition of cnoidal waves. 1.5 Hirota method In 1971, Hirota showed that certain evolution equations can be reduced to bilinear differential equations. He introduces a dependent-variable transformation (Drazin and Johnson, Hirota) w2 uxt(,) 2 ln(,) f xt , (1.5.1) wx2 wheref has the property ff,,xxx f o 0, as x of. We shall describe the method in the context of the KdV equation uuuutxxxx60 . Substituting (1.5.1) into the KdV equation we obtain 2 ffxt f x f t ff xxxx 43 f x f xxx f xx 0 . (1.5.2) This equation can be reduced to a bilinear form by using the Hirota operator (Hirota and Satsuma) mn DDtx:, Vuo V V mn wwmn ww (1.5.3) DDtx( abxt , )( , ) ( ) ( ) axtbxt ( , ) ( ', ') , wwtt'' ww xx x x ' tt ' where m, n are positive integers, V is a functions space, in particular Vf ^ : Ruou R R, f Cnm (R) C (R)` , and ab, two arbitrary functions in V . By virtue of the definition (1.5.3), the Hirota operator is 1. Bilinear mn mn mn DDtx(,)(,)(D aab12 D DDabDDabtx1 tx2,), mn mn mn DDtx(, aD b12 b ) D DDtx (, ab1 ) DDtx (, ab2 ), 26 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS Daabb1212,,, V , R. 2. Symmetric mn mn mn nm DDtx(,) ab DDtx (,) ba,,Dtx D(,) ab Dxt D (,) ab ab, V. For mn 1 , the Hirota operator (1.5.3) reduces to DDtx(,)(,) ab xt ( axt b ab tx ab xt ab tx )(,) xt , (1.5.4) and for mn 0, 4 , to 4 Dabxtabababababxtx (,)(,) (xxxx 4 xxx x 6 xx xx 4 x xxx xxxx )(,), (1.5.5) ab, V. The equation for f (1.5.2) can be rewritten in a form of a bilinear differential equation 4 DDxt(,) f f Dx (,) f f 0, (1.5.6) or, by denoting 3 BDDD x ()tx, to Bf(,) f 0. (1.5.7) To solve (1.5.7), we consider the solutions of the form N n fxt(,) ¦H fn (,) xt, (1.5.8) n 0 with Hz0 a positive, real number, and fxt0 (,) 1, (,)xt R×R. Substitution of (1.5.8) into (1.5.7) yields Bf(,)00 fH [(,1)(1,)]... Bf 1 B f1 nN (1.5.9) ...nNBf ( , f ) ... Bf ( , f ) 0. H ¦¦nr r H Nr r rr 00 From Bf(,)000 f , we obtain Bf(,1)0,1 2(Bf21 ,1) Bf ( , f1 ), (1.5.10) 2(Bf32 ,1) Bf ( , f11 ) Bf ( , f2 ), n1 2(Bf ,1) Bf ( , f ), nN2,..., , nn ¦ rr r 1 that are the sufficient condition for the equation (1.5.9) to be verified, H real and positive number. So this time, we calculate some particular expressions for B MATHEMATICAL METHODS FOR NONLINEAR EQUATIONS ANALYSIS 27 Bab(,)(,) xt ( abxt ab t x ab x t ab xt a xxxx b x 4 a xxx b x 64abxx xx ab x xxx ab xxxx )(,), xt whereab, V . Forb { 1 we find Ba(,1) axt a xxx . For axt(,) exp()T1 , T11 kx Y 1 t D 1, with k111,,YD, real and positive numbers, we have 4 Bkk(exp(),1)(T1111 Y )exp() T1 . (1.5.11) Again, for axt(,) exp()T1 , bxt( , ) exp(T2 ) , Tii kxYD i t i,1, i 2 , it follows that 4 Bkkkk[exp(TT122121211 ),exp( )] [(YYTT )( ) ( ) ]exp(2 ). (1.5.12) The expressions (1.5.11) and (1.5.12) suggest some forms to be chosen for the 3 functionsfnn , t 1 . For fxt1 (,) exp()T1 , the equation (1.5.10) 1 leads to Y11 k , and the others equations of (1.5.10) are identically verified if fnn {t0, 2 . Thus, we obtain 3 fxt11(,) 1H exp( kxkt 11 D ), (1.5.13) kkxkt23DHln uxt(,) 1111 sech[2 ]. (1.5.14) 22 An initial condition of the form ux( ,0) 2sech2 x, (1.5.15) leads to D11 ln H ,k 2 . We can say that a solution of the KdV equation with the initial condition (1.5.15) is given by the soliton wave ux( ,0) 2sech2 ( x 4 t ) . (1.5.16) If we consider the function fxt11( , ) exp(TT ) exp(2 ) , Tii kx Y i t D i,1, i 2 , with k111,,YD, positive and real numbers, the equation (1.5.10) 1 is verified if 3 Y2 kii , 1, 2 . For Bf(,)11 f we have 4 Bf(11 , f ) 2[( k 2YY k 1 )( 2 1 ) (k 2 k 1 ) ]exp( TT 1 2 ) . 28 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS Note that (1.5.10)2 suggests the form fxt21( , ) A exp(TT2 ) , where A is a real constant which is determined from (1.5.10) 2 kk12 2 A21 ()exp(TT2 ) . kk12 The constants kk12, are determined so that kk12z0 . Choosing fnn 0,t 3 , the other equations (1.5.10) are identically verified. We have 22kk12 fxt( , ) 1H [exp( T12 ) exp( T )] H ( ) exp(TT 12 ), (1.5.17) kk12 ()kk 2 Hkkexp( T ) H exp( T ) H2 12exp(T T ) w 1122kk 12 ux(,)t 2 [ 12 ], (1.5.18) wx 22kk12 1H exp( T12 ) H exp( T ) H ( ) exp(T1 T2 ) kk12 3 with Tii kx k i t D i,1, i 2 . An initial condition of the form ux( ,0) 6sech2 x , (1.5.19) 3 is verified for D D ln ,kk 2, 4 . Consequently, the solution is 12H 1 2 w6exp( 6x 72txtx ) 6exp( 2 8 ) 12exp( 4 64t ) uxt(,) 2 [ ], (1.5.20) wxxtxtx3exp(2 8)3exp(4 64) exp(6 72)t and represents the two-soliton solution. 3 For fxt( , ) exp( ) , kx t,1,2, i 3 , the equation (1.5.10) leads 1 ¦ Ti Tii Z i D i 1 i 1 3 to Zii ki,1,2,3 . 3 Choosing fxt(,) A exp( ), the equation (1.5.10) is verified for 2 ¦ ijT iT j 2 ij,1 2 1 ()kkij Aiij 2 ,,j 1,2,3 . 2(kkij ) For 3 fxt( , ) A exp( ),ijl , , 1,2,3 . 3 ¦ ijl T i T j T l ijl,, 1 MATHEMATICAL METHODS FOR NONLINEAR EQUATIONS ANALYSIS 29 The equation (1.5.10)3 yields 2 1 ()()()kkij kkkkkk iljlijl k Aiijl 2 ,,j 1,2,3. 2()()()kkij kkkkkkkiljlijl For fxtn (,) 0, nt 4, equations (1.5.10) are verified and, finally, we have 33 fxt( , ) 1 exp( )2 A exp( ) H¦¦ THiiji TTj ii 1,1j 3 3 Aiexp( ), ,j ,l 1,2,3. H¦ ijl T i T j T l ijl,, 1 In summary, we have fxt(,) 1HTTT [exp123 exp exp] 2 2 2 ()kk12 ()kk32 H [22 exp(T12 T ) exp(T32 T ) ()kk12 ()kk32 2 ()kk133 ()kk12 ()()kkkk1323 (1.5.21) 2 exp(T13 T )] H u ()kk13 ()()()kk121323 kkkk 333 kk31()()() k 2 kk 23 k 1 kk 1 2 k 3 u exp(TTT323 ). kk12 k3 Substitution of (1.5.21) into (1.5.1) leads to a solution ux(,)t , which is a three- soliton solution. An initial condition of the form ux( ,0) 12sech 2 x, (1.5.22) is verified for 61510 D ln , D ln , D ln ,kkk 2, 4, 6 . 123HHH 123 It follows that w2 uxt( , ) 2 [ln(1 6exp( 2x 8 t ) 15exp( 4 x 64 t ) wx2 10exp( 6x 216txtx ) 10exp( 6 72 ) 15exp( 8 224t ) (1.5.23) 6exp( 10xt 280 ) exp( 12 xt 288 ))]. In an inductive way, the N -soliton solution is obtained as N fxt( , ) exp( ) , kx kt3 ,1,..., i N, 1 ¦ Ti Tii D i i i 1 and the solution is 30 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS NN 2 fxt( , ) 1H¦¦ exp( THii )Ajij exp( TT ) ... ii1,1j (1.5.24) N HN A exp( T ... T ). ¦ ii11... NN i i ii1 ...N 1 We mention that the Hirota bilinear method obtains the exact N -soliton solution for many evolution equations. The bilinear method is effective for completely integrable systems and also for nearly integrable equations. This method is more useful for stability analysis than the ordinary perturbation method or the perturbation treatment of the inverse scattering method. Next, we are going to investigate the stability of the KdV soliton with respect to wavefront bending, for example. The amplitude of the wave is calculated for a small perturbation of the argument T . If the amplitude remains constant, the soliton is stable. Consider a particular solution of the KdV equation in the bilinear form (1.5.6) f 1 exp( [ 20 ) , [00 kxVt( ) . (1.5.25) Now assume the phase and amplitude of the KdV equation are slowly varying functions of y , perpendicularly to direction x . To study the stability of the soliton against transverse perturbation, it is convenient to introduce the Kadomtsev and Petviashvili equation, referred to as the K-P equation (Matsukawa et al.) (6uuuut x xxx ) x u yy 0 , (1.5.26) rewritten as a bilinear differential equation 24 ()DDxt D y D x( f, f)0 . (1.5.27) The perturbation modifies (1.5.25) as f fgˆ ˆ , fˆ 1exp(2) [, gˆ 1g , [ kx( Vt) . (1.5.28) Substituting (1.5.28) into (1.5.27), we obtain [(DD D24 D)( fˆˆ , f )] gˆˆ 22 f ˆ [( DD D 24 D)( gg ,ˆ )] xt y x xt y x (1.5.29) 22ˆˆ 6(DffDggxx ( , ))( (ˆˆ , )) 0. By expanding the expression exp(HJDDababpq )( , ) [exp( HJ DDabDDabpq )( , )][exp( HJpq )( , )] , (1.5.30) in a power series of H and G , two identities are derived 22 DDpq(,)( abab DDpq (,)) aa b a ( DDpq (,)) bb , 4422422 Dababpp(,)( Daab (,)) aDbb ( ppp (,))6((,))((,)) DaaDbb. Inserting (1.5.28) into (1.5.30), the K-P equation (1.5.27) becomes MATHEMATICAL METHODS FOR NONLINEAR EQUATIONS ANALYSIS 31 2 [8([kkxtt 0, )exp( [ 2 ) 4kt exp( [ 2 )(1 exp( [ 2 ))](1 2g ) ˆˆ24 22 2fgxt 32 k exp( [ 2 )(1 2g ) 2 fgxx 96 k exp( [ 2 ) gxx (1.5.31) [ 4(kkx )exp( [[ 2 )(1 exp( 2 )) 2fgˆ 2 0. yy k 0,yy yy The zero-th order estimation in (1.5.31) yields 2 x0,t 4k 0 , (1.5.32) where, we observe that the amplitude of the soliton is constant, as is expected. From the first order estimation in (1.5.31) we have 22 2 2 kkx(4[[[[0,ttt )sech k sech k (1 tanh ) [ (1.5.33) ggg 12 kg22 sech [ ( k kx )(1 tanh [ ) 0, xt yy xx xx yyk 0, yy where the phase and amplitude of the soliton are dependent on y . Introducing a new coordinate which is moving at soliton velocity [ kx(4 kt2 ) , (1.5.34) we observe that the perturbed solution g depends only on [ , and then equation (1.5.33) is simplified [ kg42[[[[[ 4(3sech 1)g k sech 2 ( k k kx )(1 tanh ) 0. (1.5.35) [[[[ [[ ttyyyk 0, y Multiplying (1.5.35) by 2sech 2[ and integrating with respect to [ , it results that 22 uk sech [, satisfies the KdV equation uuuutxxxx60 . From (1.5.35) we have ffk (kkx[[[[[ ) (1 tanh )dyy (1 tanh )d 0 . (1.5.36) tyy0, ³³ fk f Combining (1.5.32) with (1.5.36) the wave equation for the oscillation of the phase is obtained 2 x0,tt 4kx0,yy , (1.5.37) and yields to the conclusion that the soliton is stable against transverse perturbation. 1.6 Linear equivalence method (LEM) The linear equivalence method (LEM) was introduced by Toma to solve nonlinear differential equations with arbitrary Cauchy data. Consider the Cauchy problem 32 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS fx §·x1 §·1 ¨¸ ¨¸ dx x2 ¨ fx2 ¸ fx ,, x ¨¸ fx , (1.6.1) dt ¨¸} ¨} ¸ ¨¸ ¨¸ ¨¸x ¨¸ ©¹n ©¹fxn xt 000 =x, t I R, where f xfxQ , with the coefficients f defined on the real interval I . Here jj ¦ Q jQ Qt1 n Ȟ Q j Ȟ (Ȟ12, Ȟ ,...Ȟ n) is a multi-indices vector with n components, and z 3z j . LEM j 1 consists of applying to (1.6.1) an exponential transform depending on n parameters ȟ {[ 12,,,[[! n , which maps the equation into a linear first order partial differential equation, with respect to the independent variable n vt, exp , z , R n , , z z , (1.6.2) V V ! V V ! ¦ Vj j j 1 or vt ,VV12 , ,... Vn exp( V1122z V z ... Vnn z ) , Vi R . (1.6.3) wv wv w2v Computing , and taking account of vzi , vzij z , the result is a wt wVi wVij V linear partial differential equation wv n fDv0 , (1.6.4) [¦ jj [ wt j 1 where the formal operator from the left-hand side n , fD f D , (1.6.5) ȟ [[{[¦ jj j 1 Q Q w is obtained by replacing x to in the expression of f j x . wȟQ The Cauchy conditions (1.6.1)2 become vt 0 ,exp,ȟȟ x0 . (1.6.6) Toma has proved that the solution of the nonlinear Cauchy problem (1.6.1) is equivalent to the analytic solutions in [ of the linear problem (1.6.4) and (1.6.6). Let us consider v of the form ȟ Q vtx, v t . (1.6.7) ¦ Q Qt1 Q! MATHEMATICAL METHODS FOR NONLINEAR EQUATIONS ANALYSIS 33 The coefficients from the right-hand side of (1.6.7) satisfy a linear infinite differential equation with constant coefficients n f k vjfve jjje ,, G (1.6.8) QJQJ¦¦ j kn 1, j 11J with Cauchy conditions Q vtQ 0 x,Q N. (1.6.9) The first n components of the solution of (1.6.8) and (1.6.9) coincide with the solution of the initial problem (1.6.1). For a given vector ggx , the Lie derivative of an analytic function O is j j 1,n Lgg O{,grad O . (1.6.10) So, we can write ȟ , fD [ Lvf . (1.6.11) As a result, the following theorems hold. THEOREM 1.6.1 (Toma) The linear equation (1.6.4) is expressed in terms of Lie derivatives as wv Lv{ L v 0 . (1.6.12) wt f and the analytic solution of (1.6.4) and (1.6.6) have two expansions f k tt 0 ,1 exp, , (1.6.13) vt ȟ ¦ Lf x0 [ k 1 k! f tt 0 vt,1 ,f D exp, x . (1.6.14) ȟȟ ¦ [ 0 [ k 1 k! The important point is that the solution of (1.6.1) is written as xt grad vt ,ȟ , (1.6.15) [ ȟ 0 where v is replaced by (1.6.13) or (1.6.14). The linear infinite system (1.6.8) may be written in the matrix form dV AV,, V Vjj V vJ t , (1.6.16) dt jIN J j where the matrix A is defined as 34 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS ªºAAA11 12 13 }} } } «»0 AA «»22 23 }} } } «»00A33 }} } } A «». (1.6.17) «»}}}}} } } «»000}}AA «»jj j,1 j ¬¼«»}}}}} } } It is evident that the rectangular matrices Akjjj,1 k ,1,N!, depend only on those f jQ for which Q k . The diagonal cells Ajj depend only on the linear part of the differential operator more precisely, A11 is the associated Jacobi matrix. The Cauchy conditions (1.6.1)2 are mapped through (1.6.6) into Q Vt00 x . (1.6.18) QN The first n components of the solution of (1.6.16) and (1.6.18) coincide to solution of (1.6.1), and this holds even for systems with variable coefficients. The proof of this result is based on the construction of the inverse matrix for nonlinear operators. In the case of constant coefficients, this inverse allows a simplified exponential form and thus, the solution of (1.6.16), (1.6.18) may be written as Vt Vt 0 exp[ At t0 ] . (1.6.19) THEOREM 1.6.2 (Toma) The solution of initial problem (1.6.1) coincides with the first n components of the vector V given by (1.6.19). We mention that the block structure of A enables the step-by-step calculus of Ak . A secondary result is that the first n rows of Ak may be expressed by blocks, each of them being computable by a finite number of steps. Next, denote by Pn the operator that associates to a matrix its first n rows. Then, by theorem 1.6.2 and (1.6.19), it follows that the solution of (1.6.1) coincides with the Taylor series expansion k f tt 0 k Q xt x0 PAn x0 . (1.6.20) ¦ QN k 1 k! The above results enable a comparison with the Fliess expansions in the optimal control problems. The Fliess expansion corresponding to solution of (1.6.1) is obtained as a Taylor series f k tt 0 x tx Lx, j 1,2...,n , (1.6.21) jj 00¦ fj k 1 k! MATHEMATICAL METHODS FOR NONLINEAR EQUATIONS ANALYSIS 35 k whereLxf j0 are the k-th order iteratives of the Lie derivative Lxf j0 . Comparing this with the LEM expansion (1.6.21), we obtain the following result: k THEOREM 1.6.3 (Toma) The Lie derivative Lxfj 0 of order k with respect to f is calculated as follows k §·Lxf 10 ¨¸ Lxk ¨¸f 20 k Q PAn x0 . (1.6.22) ¨¸# QIN ¨¸ ¨¸Lxk ©¹fn 0 We have PAm S , (1.6.23) nkm kIN where SAAAAAAJJ12 Jk , (1.6.24) km ¦ 11 12 22 23}k 1, k kk J mk 1 J J12,,, J! Jk is a multi-indices vector. The calculus of Skm is easier if the eigenvalues of Akk are specified. These eigenvalues are n , , (1.6.25) LO {¦ LOj j j 1 whereO O12,,, O! On are the eigenvalues of A11 . Now we are ready to calculate the normal LEM representations, useful for a qualitative study of nonlinear equations. THEOREM 1.6.4 (Toma) The solution of (1.6.13) may be expressed as a series with respect to the data f x tx utxjj J , 1,2,...,n , (1.6.26) jj 00¦ J J 1 jj where J k and Uuk { t satisfy the linear finite systems J J k dU 1 AU , dt 11 1 dU 2 AU AU, dt 22 2 12 1 (1.6.27) }}}}}}}}}}}} dU k A UAU } AU , dt kk k k,1 k k 1 1k 1 36 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS and Cauchy - conditions j Ut10 Gms,0,2,Ut0 s ..., k. (1.6.28) mn 1, Here, the star stands for transpose matrix. This completes the description of the LEM. To illustrate its application, consider two examples. EXAMPLE 1.6.1 Let the problem be (Toma) 2 yyt , yy(0) 0 . (1.6.29) By applying the exponential transform vy exp(V ) we obtain the equivalent linear equation wwv 2v V , (1.6.30) wt wV2 and the condition vy(0,V ) exp( V0 ) . Consider vt(,V ) I\V () t ( ), and then (1.6.30) yields dI d2\ OI , V O\V(), dt dV2 with O an arbitrary constant. Next, consider f V j () , \V ¦ Mj j 0 j! to yield * j\j1 O\j , j N . We observe that \0 0 . The coefficients are determined from O j1 \ \, j N . j (1)!j 1 For vt(,V ) it results 0 f OVjj1 vt(,V ) 1 exp( Ot ) \Od , (1.6.31) ³ ¦ 1 f j 1 (1)!!jj whent t 0 . From the initial condition (1.6.29)2 it also results 0 OVjj1 \Od y j , j N . ³ 10 f (1)!!jj MATHEMATICAL METHODS FOR NONLINEAR EQUATIONS ANALYSIS 37 So, we have O \1 exp( ) , y0 and the solution becomes 0 1 f OVjj1 vt(,V ) 1 ³ exp[( Ot )]¦ dO f yjj0 j 1 (1)!! j §· f ¨¸1 V j y 1exp[¦ ¨¸ 0 V]. j 1 ¨¸1 jty!10 ¨¸t ©¹y0 Finally, this solution leads to the solution of (1.6.29) given by y yt() 0 . 1 ty0 EXAMPLE 1.6.2 The Troesch problem (Toma). Let us have the problem wntt sinh( nw ) , ww(0) 0, (1) 1 . (1.6.32) By noting x nt,() y x nw , (1.6.33) we obtain yyxx sinh , (1.6.34) with conditions yynn(0) 0, ( ) , yc(0) E. (1.6.35) The last condition (1.6.35) with a fictitious parameter E , is introduced in order to apply LEM. So, we have vx(,V[ ,) exp( V y [ yc ). (1.6.36) The linear equivalent equation is given by wwvvf 1 w21k v V [ 0 , (1.6.37) ¦ 21k ww[wVxkk 0 (2 1)! and conditions (1.6.35) become v(0,V[ , ) exp( [E ) . (1.6.38) The equivalent linear system is 38 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS dV MV , (1.6.39) dx where V and M are written as VV [],21j jN , VVikj21jik [], 21, ªºAB1 11 B 13 B 15 ... B1,2jj 1 B 1,2 1 ... «»0 AB B... B B ... «»333353,213,21jj M «»...... , «» «»0 0 0 0 ... AB21jjj 21,21 ... ¬¼«»...... Bb21,221j kj [] qs, 1,12ddqj dd s k 2 j , (1)q Gq1 b s , qs (2kj 2 1)! with G , the Kronecher symbol. The three-diagonal matrices A21j defined as ªº02j 1 0 0...0 0 0 «»10220...0j 00 «» A21j «»...... , «» «»00 00...2201j ¬¼«»00 00...0210j have the eigenvalues r(2jk 2 1), 0 dd kj 1. The initial conditions are 2121kk Vq(0,E ) [E Gq ], 0 d d 2k 1,k N . (1.6.40) Then we subtract from (1.6.39) a truncated system of equations given by dV ()m M ()mmV (), (1.6.41) dx where the finite matrices M ()m are obtained by truncation from M up to the m order, that is, up to A21m . We can show that V ()m satisfy (1.6.40), and admit the representation Vx()mm(,)E exp( AxV() ) ()m (0,)E. The equations (1.6.41) and (1.6.40) are solved by the block partitioning method. The solution is obtained as follows MATHEMATICAL METHODS FOR NONLINEAR EQUATIONS ANALYSIS 39 exp(x ) 1(0) yc exp(x ) 32 yx()# yc (0) 2ln , (1.6.42) 4 exp(x ) 1(0) yc 32 where uy c(0)exp n satisfies u 1 u 32 22ln . 2 u 1 32 1.7 Bäcklund transformation The Bäcklund transformation was developed in 1883, arising from the pseudospherical surfaces construction. The name of Albert Victor Bäcklund (1845– 1922) is associated with the transformation of surfaces that bears his name, the extensions of which have a great impact in soliton theory (Coley et al.). In 1885 Bianchi shows that the Bäcklund transformation is related to an elegant invariance of the sine- Gordon equation. The geometric origins of the Bäcklund and Darboux transformations and their applications in modern soliton theory represent the subject of the monograph of Rogers and Schief (2002). One of the simplest Bäcklund transformations are the Cauchy–Riemann relations uvx y , uy vx , (1.7.1) for Laplace equations uuxx yy 0, vvxx yy 0. (1.7.2) If vxy(, ) xy is a solution of the Laplace equation, then another solution of the 1 Laplace equation uxy(, ) ( x22 y ), can be determined from 2 uxx , uyy . (1.7.3) Consider next, the Liouville–Tzitzeica equation uuxt exp , (1.7.4) and let us introduce another equation, simple to be solved vxt 0 . (1.7.5) Suppose we have a pair of relations that relate the solutions of (1.7.4) and (1.7.5) uv uv uv 2exp , uv 2exp . (1.7.6) xx 2 tt 2 40 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS The important point about (1.7.6) is the possibility to determine a solution of (1.7.4) for a specified solution of (1.7.5). Indeed, we obtain from (1.7.6) 1 uv uvxt xt ()expexp uvt t u , 2 2 1 uv uvtx tx ()expex uvx x pu , 2 2 which lead to (1.7.4) and (1.7.5). Therefore, we have the following definition. DEFINITION 1.7.1 Let us have two uncoupled partial differential equations Puxt((,))0 and Qvxt((,)) 0, where P and Q are nonlinear operators. The pair of relations Ruvuvuvxtixxtt(,, , , ;,) 0, i 1, 2, is a Bäcklund transformation if it is integrable for v when Pu() 0 and if the resulting v is a solution ofQv() 0 , and vice versa. If PQ , so that vu and satisfy the same equation, then Ri 0 , i 1, 2, is called an auto-Bäcklund transformation. The Bäcklund transformation reduces the integration of a nonlinear partial differential equation to the solutions of an ordinary differential equation, in general of low order. The existence of the Bäcklund transformation implies that there is a relation between the solutions of PQ and . The relations (1.7.6) are therefore the Bäcklund transformation of equations (1.7.4) and (1.7.5). In the following we will present the original Bäcklund transformation for the sine- Gordon equation (Rogers and Schief 2002). Let rrxyz (, ,) denote the position vector of a point P on surface 6 in R3 , written under the Monge form rxeyezxye 12(, )3 . (1.7.7) The first and second fundamental forms are defined as 2222 22 I Exd2ddd Fxy Gy (1)d2dd(1)d zxx x zzy xy z y y, 221 2 2 IIex d2ddd fxygy (d2dddzxxx zxyzyxy yy ). (1.7.8) 22 1zzxy The mean and Gaussian (total) curvature of 6 are written as 22 2 (1zzx )yy 2 zzz x y xy (1 zzy ) xx + EG F 223/2 , (1.7.9a) 2(1zzxy ) MATHEMATICAL METHODS FOR NONLINEAR EQUATIONS ANALYSIS 41 2 2 eg f zzxxyy z xy . 22 22. (1.7.9b) EG F(1 zxy z ) If 6 is a hyperbolic surface, then total curvature is negative and the asymptotic lines on6 may be taken as parametric curves. Then eg 0 . And the angle Z between the parametric lines is such that F H cos Z , sin Z . (1.7.10) EG EG 1 In the particular case when K is a constant, 6 is a pseudospherical surface. a2 If6 is parametrized by arc length along asymptotic lines corresponding to the transformation ddx oxExc ()dx , ddy oyGyc ()dy , the fundamental forms become, dropping the prime I d2cosdddxxy2 Z y2 , (1.7.11a) 2 II sinZ dxy d . (1.7.11b) a The Liouville representation of K in terms of E , G and F is given by 22 1 ª§·§HH**11 12 ·º K «¨¸¨¸» , (1.7.12) HE¬©¹© E¹¼ i where * jk are the Christoffel symbols. If 6 is a pseudospherical surface, the equation (1.7.12) is reduced to the sine-Gordon equation 1 Z sin Z. (1.7.13) DE a2 The original Bäcklund transformation is related to the pseudospherical surfaces construction with the Gaussian curvature 1 . . a2 Ifr is the position vector of a pseudospherical surface 6 , then a new pseudospherical surface 6c with the position vector rc and having the same curvature is written as rrannnnccc u,c os] , (1.7.14) where ] is the angle between the unit normals n and nc to 6 and 6c , and it is constant because ||rrc La sin] . 42 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS Mention that r 6rc is tangent to both and 6c . The angle ] is related to the 1 Bäcklund parameter P by the relation P tan ] . We write (1.7.14) in terms of 2 asymptotic coordinates L ZZcc ZZ rrc [sinr sin r ], (1.7.15) sinZ 2 DE2 where Z and Zc are related §·ZZcc P ZZ ¨¸ sin , ©¹22D a §·ZZcc1 ZZ ¨¸ sin . (1.7.16) ©¹22E Pa The relations (1.7.16) represent the standard form of the Bäcklund transformation of the sine-Gordon equation. Here, Z and Zc are solutions of the sine-Gordon equation. § ZZc · § ZZc · 1 Indeed, by calculating ¨ ¸ and ¨ ¸ , we have ZDE 2 sin Z, and ©¹2 DE ©¹2 ED a 1 Zc sin Zc . Since both Z and Zc are solutions of the sine-Gordon equations DE a2 (1.7.16) is an auto-Bäcklund transformation for (1.7.13). If Z 0 , the transformation (1.7.13) becomes §·ZPZcc§·ZZcc1 ¨¸ sin , ¨¸ sin . (1.7.17) ©¹22D a ©¹22E Pa Integrating we obtain PZZc d1c D 2log|tan Zc | f (E ) , (1.7.18) ³ Zc a sin 4 2 and 11 E 2log|tan Zc | g (D ), (1.7.19) Pa 4 wheref and g are arbitrary functions. From (1.7.18) and (1.7.19) we obtain the kink solution of the sine-Gordon equation 1 P 1 tanZc C exp( D E ) , (1.7.20) 4 aaP or P 1 ZDEc( , ) 4arctan{C exp(D E )} , (1.7.21) aaP MATHEMATICAL METHODS FOR NONLINEAR EQUATIONS ANALYSIS 43 with C a constant. The Bäcklund transformation admits the nonlinear superposition principle, namely, if we refer to (1.7.13) ZZPP ZZ tan 12 2 1tan 2 1 . (1.7.22) 44PP21 IfZ1 and Z2 are two solutions for the sine-Gordon equation, generated from Z by means of Bäcklund transfomation (1.7.16) of parameters P1 and P2 , then Z12 given by (1.7.21) is a new solution for the sine-Gordon equation, that is a two-soliton solution. Writing ZoZ into (1.7.21) we have ZZPP ZZ tan 12 2 1tan 2 1 . (1.7.23) 44PP21 Relations (1.7.22) and (1.7.23) are known as permutability theorem of Bianchi. Any proof of permutability theorem is reduced to equality Z12 Z 21 . Luigi Bianchi established in 1892 that the Bäcklund transformation for the sine-Gordon equation admits a commutative property. This property leads to construction of nonlinear superposition formulae for solutions of evolution equations. The Bianchi theorem may be interpreted as an integrable discrete equation, if we assume that Z is a point into a discrete lattice of axes n1 and n2 Z Z(,nn12 ), (1.7.24) and the solutions Z1 ,Z2 and Z12 are neighbor points in this lattice (Rogers and Schief 1997) Z11 Z(1,nn 2 ) , Z212 Z(,nn 1), (1.7.25) Z12 Z(1,1nn 1 2 ) . (1.7.26) Therefore, the Bäcklund transformation (1.7.16) can generate discrete pseudospherical surfaces 6 , associated to the sine-Gordon equation. Let us illustrate the application of the Bäcklund transformation with some examples (Lamb, Drazin). EXAMPLE 1.7.1 (Lamb) A general class of soluble nonlinear evolution equations are derived from linear equations. Let us consider the linear equations vvq11x i] v2 , vvq22x i] v1 , (1.7.27) whereq must be determined, and introduce a set of additional equations ww11x i] Mw2 , ww22x i] Mw1 , (1.7.28) whereM must be determined. Suppose that v and w are related by 44 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS §wv11 ·§· §· v1 ¨ ¸¨¸ ¨¸, (1.7.29) ©wv22 ¹©¹ ©¹ v2 where is a 22u matrix of elements aij . Taking account of (1.7.27), we rewrite (1.7.29) under the form wAvBv11 2 , wCvDv21 2 , (1.7.30) where Aa 11 i] , Ba 12 q, Ca 21 q , Da 22 i] . By choosing aa21 12 , so that C B , and substituting (1.7.30) into (1.7.28) we obtain some relations between q and M Ax ()qBM, BBqADx 2i] M, BBBqx 2i] M D , DqBx ()M . (1.7.31) These equations admit first integrals ADB22 ft() and A Dgt 2(), that imply BfgAg222 ()2 . So, the relation ()2(ADx Bq M) can be written as A x q M. (1.7.32) hAg22() Denoting qz x , M zxc and integrating, we obtain A ghcos( zz c ) , Bh sin( zzc ) , Dghzz cos( c ) . (1.7.33) The integration constant has been chosen so that Bzz vanishes when and c vanish. 1 Next, we set for simplicity gh , and obtain from (1.7.31) 2 ()2isin(zzccx ] zz ) . (1.7.34) The equation (1.7.34) relates zz and c , and is a Bäcklund transformation. The second Bäcklund transformation, which relates zt and ztc , depends upon the particular evolution equation being considered. According to the permutability theorem, by introducing zzo zc ozc into (1.7.34), we obtain zz aa z z tan(30 ) r 12 tan( 21 ) , (1.7.35) zaa12 2 where ai 2i]i . Applying (1.7.35) to all evolution equations for which (1.7.27) are the appropriate linear equations, we can calculate another solution qz x . MATHEMATICAL METHODS FOR NONLINEAR EQUATIONS ANALYSIS 45 EXAMPLE 1.7.2 Let us consider the KdV equation Pu() utxxxx 6 uu u 0. We introduce a new dependent variable w , wux , and define the operator 2 Qw() wt 3 w x w xxx , so that (())Qwx Pu (). The Bäcklund transformation is given by 1 wwcc 2(O ww)2 , (1.7.36) xx 2 2 2 wwt tccc ()()2( wwwxx w xx u uuu c c ) . (1.7.37) Suppose that w1 and w2 are two solutions that verify (1.7.36) and (1.7.37), and w0 is a solution of Qw() 0 , that is 1 ww O2( ww)2 , (1.7.38a) 10xx 12 10 1 ww O2( ww)2 . (1.7.38b) 20xx 22 20 Similarly, we can construct a solution w12 from w1 and O2 , and another solution w21 from w2 and O1 1 ww O2( ww)2 , (1.7.38c) 12xx 1 22 12 1 1 ww O2( ww)2 . (1.7.38d) 21xx 2 12 21 2 From the Bianchi permutability theorem it results ww12 21 . Then, subtract equations (1.7.38a) and (1.7.38b), and similar equations (1.7.38c) and (1.7.38d), and add the results to obtain 111 1 04()( OOww )22 ( ww ) ( ww ) 2 ( ww )2 , 12222 10 2 0 121 2 122 or 4(OO12 ) ww12 0 . ww12 46 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS Therefore, this method is remarkable for generation of new solutions. In particular, for w0 0 , O1 1 , wxt1 2tanh( 4 ), O2 4 , and wxt2 4coth(2 32 ) , we have 12 wxt(,) , 12 [2coth(2x 32tx ) tanh( 4t )] that yields to a two-soliton solution 3 4cosh(2x 8tx ) cosh(4 64t ) uxt(,) 12 . 12 [3cosh(x 28txt ) cosh(3 36 )]2 1.8 Painlevé analysis Before discussing the Painlevé property of an ordinary differential equation, we give a short review of singular points and their classification (Gromak). Let us consider a nonlinear differential equation in the complex plane wfzwc (, ), (1.8.1) wherefD: o Cn1 Cn is an analytic function of complex variable. A solution of (1.8.1) is an analytic function, which is determined by its singular points. Here dw wc . dz DEFINITION 1.8.1 A point at which wz() fails to be analytic is called a singular point or singularity of wz() . Singular points can belong to the following classes: a) Isolated singular points. b) Nonisolated singular points. c) Single-valued points, for which the function does not change its value as z goes around a given initial point z0 . d) Multi-valued or branch, or critical points. e) The points for which the function has a limit, whether finite or infinite, as zzo 0 . f) The points for which the function wz() has a limit as zzo 0 , namely z0 is an essential singular point. We recall that a critical point is a singular point at which the solution is not analytic, which is not a pole. MATHEMATICAL METHODS FOR NONLINEAR EQUATIONS ANALYSIS 47 PROPOSITION 1.8.1. Consider the system 0 wfzwwwcj jn( ,12 , ,..., ) , wzj ()0 wj , jn 1,2,..., , (1.8.2) 0 and let f be analytic in the domain ||zzd0 a, ||wwjj d b and M be the upper bound of the set f in this domain. Then the system admits a unique solution wz() , which is analytic within the circle| zzdU0 | and which reduces to w0 , whenzzo 0 , b U a[1 ex p ] . In the linear case U a . Mn(1 ) A singular point may have more than one of the above properties. If (,zw00 ) is a singularity of f then wz() can have a singularity at z0 . For example, the equation zwc w 0 , admits the general solution c wz() , z wherec is an arbitrary constant. The solution has a simple pole z 0 for c z 0 . The nonlinear equations may have movable singularities, whose position does depend on the arbitrary constants of integration. The equation wwc 2 0 , admits the general solution 1 wz() , zz 0 wherez0 is an arbitrary complex constant, which is a simple movable pole. The equation dw exp(w ) , dz has the general solution wz() log( z z0 ), wherez0 is an arbitrary complex constant. This solution has a logarithmic branch point (critical point) at the movable point zz 0 . DEFINITION 1.8.2 The movable singularities of the solution are the singularities whose location depends on the constant of integration. Fixed singularities occur at points where the coefficients of equation are singular. DEFINITION 1.8.3 An ordinary differential equation is said to possess the Painlevé property when all movable singularities are single-valued (poles), that is when 48 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS solutions are free from movable critical points but can have fixed multivalued singularities. Consider in the following those equations which do not contain movable singular points. In 1884, Fuchs shows that if the first order equation (Drazin and Johnson) wFzwc (, ), where F is a rational function in w , and analytic in z , does not contain any movable crtical points, then wc Fzw(, ) az () bzw () czw () 2 , (1.8.3) with abc,, analytic functions. The above equation is the Riccati equation. Painlevé and Gambier (see Gromak) extended these ideas to equations of the second order wFzwwcc (, , ), and showed that there are only 50 equations which have the property of having no movable critical points. They showed that 44 equations were integrable in terms of known functions, such as elliptic functions and solutions of linear equations, or were reductible to one of six new nonlinear differential equations, namely (Gromak) wwcc 6 2 z , wzwwcc 2 3 D, 111 G wwwcc c22DEJ c () w w3 , wzz w 13 E wwwzwzwcc c 23D42() 22 , 22ww 31wwE 1 1 JG(1w ) wwwwwcc c22D c (1)( ) , 2(ww 1) z z2 w z w 1 11 1 1 1 1 1 wwcc ()(c 2 )wc 21w w wz z w 1 wz ww(1)() w z E z JG(1)(1) z zz [] D , zz22(1) ( w 1)( 2 w 1)( 2 wz ) 2 with DEJ,, and G arbitrary constants. These equations are known as the Painlevé equations. The first three equations were discovered by Painlevé, and the next three by Gambier, Ablowitz, Ramani and Segur expressed in 1980, the sufficient and necessary conditions of integrability for a nonlinear partial differential equation; every ordinary differential equation derived from it by exact reduction, must satisfy the Painlevé property. Each of the above equations can be analyzed in terms of meromorphic functions theory. In particular, let us consider the second Painlevé equation MATHEMATICAL METHODS FOR NONLINEAR EQUATIONS ANALYSIS 49 wzwwcc 2 3 D, (1.8.4) which is a meromorphic function of z. In the neighborhood of each pole z0 , the solution has the expansion (Gromak) HH DH 232 w zzz00( ) ( zz0 ) hzz ( 0 ) ..... , H 1, (1.8.5) zz 0 64 whereh is an arbitrary constant. The theorem of meromorphic functions implies that each solution of (1.8.4) can be expressed as the ratio of two entire functions, that is vz() wz() . (1.8.6) uz() The expansion (1.8.5) leads to 2 1 z0 w 2 Oz( z0 ) , (1.8.7) ()zz 0 3 and z 1 wz2d( Ozz ) . (1.8.8) ³ zz 0 z1 0 Therefore the function §·zz uz() exp¨ d z w2 d z¸ , (1.8.9) ¨ ³³ ¸ ©¹zz11 has a simple zero at z0 , and it is entire. The function vz() is also entire. From (1.8.6) it follows the equations for uz() and vz () uucc u c 2 v2 , (1.8.10) vucc22 vu c2 uvu c c v 3D zvu 2 u3 . (1.8.11) When v is expressed as a polynomial in u , the equation (1.8.10) reduces to a Weierstrass equation (1.4.14). It can be proved that all the solutions of (1.8.9) are entire, and, if (,uv)0,u z is an arbitrary solution of (1.8.9), then (1.8.6) is a solution of (1.8.4). The equation (1.8.3) has a one-parameter family of solutions, which is defined by the general solution of Riccati equation (1.8.3). Indeed from this equation we obtain wawbwccc c22 c c (2 awbawbwc )( ) , z a and the coefficients abc,, are obtained, a2 1, b 0 , c , D . 2a 2 By employing the Painlevé analysis, the explicit solutions for several nonlinear equations can be obtained. To illustrate this, we present further some examples of Painlevé analysis application. 50 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS EXAMPLE 1.8.1 (Satsuma) Exact solutions of a nonlinear reaction–diffusion equation of the type 2 uutx ()x Fu (). (1.8.12) In (1.8.12) F()u is a polynomial. In population dynamics, this equation is a model for the spatial diffusion of biological populations. Let us consider the case F()uuu (D 1)( u ). (1.8.13) Substituting uuzuxct () ( ) into (1.8.12), we have 2 ()ucuuuuzzD z ( 1)( )0 . (1.8.14) We suppose that the solution is a meromorphic function which can be written as a Laurent series, that is f uz() a ( z z )n . (1.8.15) ¦ n 0 nm For simplicity we take z0 0 . Substitution of (1.8.15) into (1.8.14) and equating the 5 1 terms with the same power in z , yield to m 2 , a (1D ) , ac , a 0 , 0 12 1 28 1 a2 20 , and so on. It is easy to verify that the determination of a4 leads to c 0 . For this particular value of c , the solution (1.8.15) becomes 515 uz 202 DDD (1 ) { (1 )22 } z ... . (1.8.16) 12 60 16 Analyzing the nature of the singularities in (1.8.16), we choose for u the form f u , g fAB exp( kzC ) exp(2 kz ) , (1.8.17) g [1 exp(kz )]2 , where K,,AB and C are constants. Substituting (1.8.17) into (1.8.14) we find two nontrivial cases: 1 3 Case 1. A 1, B 2 , C 1, k , D . It follows that 5 5 2 1 °tanhzz , 0, u ® 25 (1.8.18) ° ¯ 0, z t 0. MATHEMATICAL METHODS FOR NONLINEAR EQUATIONS ANALYSIS 51 1 Case 2. A 0 , B 5 , C 0 , k i , D 0 . It results 2 51 u sec2 z . (1.8.19) 44 Solution (1.8.18) corresponds to an equilibrium solution. The equilibrium solution has the property limzzo0 u 0 , limzzzo0 u const. By setting z o iz , the solution (1.8.18) can be written as a soliton 51 u sech2 z . (1.8.20) 44 EXAMPLE 1.8.2 (Satsuma) Let us consider the equation 3 uu ()2 16( uu ). (1.8.21) txx 4 By expressing the solution u into a Laurent series, the same procedure as above, yields to an equilibrium solution 2 S °cosxx , | | , u ® 2 (1.8.22) ¯° 0, |x |t 0. Suppose that uft ()cos2 x , and substituting it into (1.8.21) we obtain 1 2 S cosxx , | | , °1 exp(12(tt )) 2 u ® 0 (1.8.23) S ° 0, |x |t , ¯° 2 for ux(0) !1, and 1 2 S cosxx , | | , °1 exp(12(tt )) 2 u ® 0 (1.8.24) S ° 0, |x |t , ¯° 2 forux(0) 1 . The solution (1.8.23) blows up at t t0 , and the solution (1.8.24) tends to zero as t of . This result implies that the equilibrium solution (1.8.22) is unstable. EXAMPLE 1.8.3 (Satsuma) For the equation 222 uutx ()x 6( uu 1), (1.8.25) 52 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS the Painlevé analysis leads to a traveling wave solution tanh(2tx ), x 2 t , u ® (1.8.26) ¯0, x t 2t . This wave is a kink, which propagates in the positive direction of the axis x . From the symmetry of the equation, the solution tanh(2tx ), x! 2 t , u ® (1.8.27) ¯0, x d 2t , is a traveling wave which propagates in the negative direction. EXAMPLE 1.8.4 (Satsuma) For equation 2 uutx ()xD u, (1.8.28) the Painlevé analysis leads to DDexp(t ) 22/3 1/3 ° [xc (1exp())],|| D t x c (1exp()), D t u ®12(1D exp(t )) (1.8.29) ° 1/3 ¯ 0, |xc |tD (1 exp( t )) , forD 0 , with c a positive constant. The solution is meaningful for t ! 0 . The amplitude monotonically decreases to zero, and the width increases to 2 c . For D!0 , two types of solutions are derived DDexp(t ) 22/3 1/3 °[xc (1exp())],||D t x c (1exp()),D t u ® 12(1D exp(t )) (1.8.30) ° 1/3 ¯ 0, |xc |tD (1 exp( t )) , DDexp(t ) 22/3 1/3 °D[xc (exp( t ) D 1) ], | x | c (exp( t ) 1) , u ® 12(exp(Dt ) 1) (1.8.31) ° 1/3 ¯ 0, |xc |tD (exp( t ) 1) , wherec are cc are constants. The amplitude of the solution (1.8.30) monotonically increases to infinity, and the ln 3/ 2 amplitude of the solution (1.8.31) admits a minimum for t , and then increases D to infinity. Chapter 2 SOME PROPERTIES OF NONLINEAR EQUATIONS 2.1 Scope of the chapter The stability of solitons is explained by the existence of infinitely many conservation laws. The conserved geometric features of solitons are related also to the symmetries. A symmetry group of an equation consists of variable transformations that leave the equation invariant. In this chapter we summarize some of the elementary principles of linear and nonlinear evolution equations, including the symmetries and conservation laws. In the classical Sophus Lie theory, the symmetry groups consist of geometric symmetries, which are transformations of independent and dependent variables. For example, the KdV equation has four such linear independent symmetries, namely arbitrary translations in the space and time coordinates, the Galilean boost and the scaling. The theorem of Emmy Noether (1918) gives a one-by-one correspondence between symmetry groups and conservation laws for Euler–Lagrange equations. The generalized symmetries introduced by Noether, are groups whose infinitesimal generators depend not only on the independent and dependent variables, but also the derivatives of the dependent variables. The generalized symmetries are able to explain the existence of infinitely many conservation laws for a given nonlinear evolution equation. Many authors have studied the properties of nonlinear equations with solitonic behavior. We refer to the monographs of Dodd et al. (1982), Teodorescu and Nicorovici-Porumbaru (1985) and Engelbrecht (1991) and to works of Wang (1998) and Bâlă (1999). 2.2 General properties of the linear waves The soliton, described by the hyperbolic secant shape, is a localized disturbance with non-oscillatory motion, having the velocity dependent of its amplitude. This contrasts strongly with the linear waves for which the velocity is independent of the amplitude. Let us consider the one-dimensional string motion equation 2 ucttu xx 0 , (2.2.1) INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 54 where c is a real positive number. Let x range from f to f . For the transverse T vibrations of a string c2 , where T is the constant tension and O , the mass per unit O length at the position x . For the compressional vibrations of an isotropic elastic solid in which the density and elastic constants are functions of x only (laminated medium), we O2 P have c2 . For the transverse vibrations of such laminated solid it follows that U P dx c2 . The characteristics are given by rc , namely the straight lines inclined to U dt the axis at c tan M. D’Alembert solution of (2.2.1) is written as ux(,)tfxctgxct ( ) ( ). (2.2.2) Functions fg,:RRo are determined from the initial conditions attached to (2.2.1) ux(,0) ) (), x ut (,0) x < () x . (2.2.3) Thus, we have 11x a fx() ) () x ³ 11x a gx() ) () x ³ A section through this surface in the plane t t0 , is uuxt (,0 ), and represents the profile of the vibrating string (a wave) at the time tt 0 . A section through the surface in the plane x x0 , is uu (,)xt0 , and represents the motion phenomenon of the point x0 . The modified profiles of fx( ct) can be determined in the following way. Consider one observer with a system of coordinates (',')x t so that, at the time t0 0 , the observer occupies the position x 0 , and at the time t , the position ct , since it has a rectilinear motion with the velocity c . In the new coordinate system (',')x t , attached to the observer ('ttxxct , ' ), the function fx( ct) is specified, at any time t '('), by fx. The observer sees, at any moment of time the unchanged profile fx() at the initial time tt00 ' 0) . This is the way the function fx( ct represents a right traveling wave or a forward-going wave with the velocity c. For a similar reason, g()xct represents a left traveling wave or a backward-going wave with the velocity c . As a consequence, both waves are not interacting between them and do not change their shape during the propagation. These waves can be called solitary waves, for the reason 55 SOME PROPERTIES OF NONLINEAR EQUATIONS they are not changing their shape during propagation process, and do not interact one with the other These waves can be superposed by a simple sum, because of the linearity of (2.2.1). The most elementary linear wave is a harmonic wave vxt(,) A expi( kxZ t ), (2.2.6) and represents the solution of (2.2.1). 2SZ The real number k is the wave number, related to the wavelength by k , O c Z is the angular frequency related to the frequency by Z 2 Sf , and the real number A is the amplitude. The phase velocity is the speed of the phase M Ztkx , and represents the velocity of propagation of a surface with constant phase Z c . (2.2.7) p k The group velocity is the speed of the bulk of the wave dZ c . (2.2.8) g dk Introducing (2.2.6) into (2.2.1) we obtain the dispersion relation, that is a relation between k and Z Fk(,)Z 0. (2.2.9) In particular, for homogeneous and isotropic media the classical one-dimensional longitudinal wave propagation equation is (2)O P Ixxtt UI , (2.2.10) whereI is displacement, OP, the Lamé elastic constants, and U density. The dispersion relation is written as k 2 (2)O P ZU2 . (2.2.11) Z Here, const. , and we say that the waves are not dispersive. If Z is real, we say k we have dispersion of waves, if the phase velocity depends on the wavenumber. For example, in the case of the linearised KdV equation uut x u xxx 0 , (2.2.12) the dispersion relation is Zkk 3 0 . (2.2.13) Z Z For Re const. , the waves are nondispersive, and for Im 0 , the waves are k k 2 nondissipative. The phase velocity cp 1 k , depends on the wavenumber and then INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 56 2 we have the dispersion phenomenon. The group velocity cg 13 k differs by the phase velocity for k z 0 , and in consequence, the components of waves scatter and disperse in the propagation process. The linear Klein–Gordon equation 2 Ixx I tt m I , (2.2.14) yields dispersion relation Z22 mk 2 . (2.2.15) If Z is complex Z Re Z i ImZZ , Im 0 , we say we have dissipation of waves. The solution in this case is vxt(,) A exp(Im)expi[ tZ kx Re(Zt )] , (2.2.16) and the amplitude is exponential decreasing at t of . For the linearised Burgers equation uutxxxu 0 , (2.2.17) the dispersion relation is Z kk i 2 . (2.2.18) The phase velocity of the harmonic waves vxt(,)exp() Atk2 expi(kx t) is cp 1 , and the group velocity, ckg 12i . The dissipation appears because Im Z k 2 is negative for any real k . In conclusion, the KdV equation is dispersive due to the term uxxx , and Burgers equation is dissipative due to uxx . Consider next the linearised SchrCdinger equation Itxx GI . (2.2.19) IfG i , (2.2.19) becomes iItxx I . (2.2.20) 2 It follows that Z k , ckp , ckg 2 , and, as we mentioned before, this is a purely dispersive equation. If G is real and positive, we obtain Z Gi k 2 , and the real part of Z vanishes. We do not have dispersion, and the waves decay as exp()Gkt2 when t of. In this case (2.2.19) is the heat equation, and is purely dissipative. Coming back to (2.2.19), let us consider the initial conditions I(,0)x fx(), with a nonharmonic function fx( )( . In this case we represent I x,0) by a Fourier integral 1 f I(x ,0) ³ Ak ( )exp(i kx )dk , (2.2.21) 2S f where 57 SOME PROPERTIES OF NONLINEAR EQUATIONS 1 f A()kx ³ I (,0)exp(i kx )d x . (2.2.22) 2S f A solution for (2.2.19) is given by 1 f I(x ,tAkkxkt ) ³ ( )exp{i[Z ( ) ]}dk , (2.2.23) 2S f with the amplitude A(k) determined from the initial conditions I(,0)x . The solution (2.2.23) can be written under the form (Dodd et al.). ff 1 ªº2 I(x ,tk ) ³³ d«» I[[[ ( ,0)exp( i kkx )d exp[i( G ikt )] , (2.2.24) 2S f¬¼ f or ff 1 ªº2 I(xt , ) «» exp{i[(x[ ) k G kt ]}d k I ( [ ,0)d [. (2.2.25) 2 ³³ S f¬¼ f Noting K x [, (2.2.25) reduces to the evaluation of two integrals f f I coskktKG exp(2 )dk , I sinkktKG exp(2 )dk . (2.2.26) 1 ³ 2 ³ f f The integrant of I2 is odd, and then I2 = 0. K We define O2 Gkt2 and a , so that Gt 1 f Ia() cos aOO exp(2 )dO . (2.2.27) 1 ³ Gt f Differentiating (2.2.27) with respect to a , and integrating by parts with respect to O , yield dI 1 S 1 aI,(0) I , (2.2.28) d2at11 G S 1 I exp(a2 ) , (2.2.29) 1 Gt 4 f where ³ exp(O2 )d O S . Finally, the solution is f 1 f ()x [ 2 I(,)xt ³ I[( ,0) exp[ ] . (2.2.30) 4SGt f 4Gt The integral (2.2.30) can be evaluated in a particular case, namely I(,0)x is a Gaussian function, I(x ,0) A exp( Dx2 ) . The result is INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 58 A Dx2 I(,)xt exp[ ]. (2.2.31) 14SDGt 14SGt IfG is real and positive, then Io0 as t of, the equation being dissipative. If G i , then I is an oscillatory function, and the initial wave is dispersed. Finally, we may say that the cnoidal waves, expressed by Jacobi elliptic functions, form a special class of nonlinear waves that bridge the linear waves to solitons. Such a class of waves is represented by solutions of the KdV equation uuuutxxxx6 0 . (2.2.32) Introducing the new variable T x c W, equation KdV becomes uuccc 6(1)2 u . (2.2.33) Multiplying (2.2.33) by 2uc and integrating we obtain a Weierstrass equation (1.4.8) uucuc23 4(1) 2 C , (2.2.34) 32 where C is a constant. For gg23!27 0 and C 0 , we obtain the solution 2 ()ue 313 ( ee )cn(*,) um, (2.2.35) ee2 3 where m and uee* 13u , with eee12t!3 , the roots of cubic polynomials ee13 from the right-hand side of (2.2.34). Coming back to the KdV equation, the solution becomes 11 ucc()T cn{2 (TT )}. (2.2.36) 22 0 These constants c and T0 are determined from the initial conditions uuu(0) 0 ,c (0) c0 . The solution (2.2.36) is called a cnoidal wave. In the limit, when u and its derivatives tend to zero at infinity, we have 11 cn22 {ccm (TT )} o sech { (TT )}, o 1, (2.2.37) 2200 and the cnoidal wave transforms into a soliton. On the other hand, for small amplitude waves, when the linearised version of the KdV equation is appropriate, we have in the limit m o 0 11 cn22 {cc (TT )} o cos { ( TT )},m o 0, (2.2.38) 2200 and the cnoidal wave transforms into a cosine oscillation. 59 SOME PROPERTIES OF NONLINEAR EQUATIONS 2.3 Some properties of nonlinear equations In nature, each effect admits an opposite, that means a contrary effect. The opposite of dispersion is the energy concentration due to nonlinearities of the medium, which manifests as a focusing of waves (Munteanu and Donescu, Munteanu and Boútină, Whitham 1974). The opposite of dissipation is amplification. The amplification of waves arises from an influx of energy in active media, where the energy is pumping from a source to wave motion, or due to some interactions between waves and medium (Chiroiu et al. 2001a). The mechanism, called the dilaton mechanism, is recently proposed for explaining the possible amplification of nonlinear seismic waves (Engelbrecht and Khamidullin, Engelbrecht). Zhurkov and Petrov introduce the dilaton concept to explain the fracture of solids. The dilaton is a fluctuation of internal energy of a medium with loosened bonds between its structural elements. The dilaton is able to absorb energy from the surrounding medium, and when the accumulated energy in it has reached its critical value, the dilaton breaks up releasing the stored energy, causing the amplification of waves. This mechanism is controlled by the intensity of the propagating wave. Low- intensity waves give a part of their energy away to the dilatons, and high-intensity waves cause the dilatons to break up. The dilatons may be distributed in a medium according to a certain order, may be absent in some regions, or may be randomly distributed. Sadovski and Nikolaev have shown in 1982 that the phase transitions in a multiphased medium can activate the dilatonic mechanism of energy pumping from a dilaton to wave motion. We can say also, that the tectonic stress concentrations in solids may grow until the stress field components reach their critical values at the points of their maximum concentrations, causing a seismically active event with the releasing of energy by the dilatons broken (Kozák and Šileny). Here is an example based on the description of long seismic waves in an elastic layer of thickness h resting on an elastic halfspace. The motion equation of a continuous medium is (Eringen 1970) ()()VUKLxfA k,, L K 0 k k 0 , (2.3.1) whereU0 is the density in underformed case, VKL is the Kirchhoff stress tensor, fk the body force, Ak the acceleration, and xk the Eulerian coordinates. The constitutive laws are given by wE V V(,E KL ), (2.3.2) KL KL KL wt ffEkkKL ( ) , (2.3.3) where EKL is the Green deformation tensor. The constitutive law (2.3.3) is referred to the body forces that result from long-range effects, and depend on EKL . Noting with U the transverse displacement, the motion equation of waves in the positive direction of the axis X is obtained from (2.3.2) (Engelbrecht) INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 60 ww22UU www UUU2 4 ccmcl2222() 2 f , (2.3.4) wwtX22ss22wX w X2s20 w X 4 where X is directed along the layer, m is a nonlinear material parameter expressed in terms of second, third and fourth order elastic moduli, Qi , i 1, 2 are Poisson ratio in the layer ( i 1) and the halfspace ( i 2 ), and cisi , 1, 2 , are the velocity of transverse waves in layer and in the halfspace 2 2 22224 22cs2 2Q1 Z cks20() lk, lh0 (1)22!0 . (2.3.5) cs12Q In (2.3.5) Z is the angular frequency and k the wave number. The equation (2.3.4) may be written under the form 3 wwwum2 u11l0 u 22uf3 0 , (2.3.6) wW22Hwccss22 w[2 H [H wU where [ ct X, W H2 X , u . The small parameter H is related to the absolute s2 wt wU value of the deformation 1 . wX The constitutive law (2.3.3) is written under the form 2 3 fbUbUbU ( 123XX X ) , (2.3.7) with bii ,1,2, 3 , positive constants. wU Since the motion equation (2.3.6) is written in terms of u , we express also wt (2.3.7) under the form 2 3 fBuBuBu 12 3, (2.3.8) where b1 b2 b3 B1 , B2 2 , B3 3 . cs2 cs2 cs2 Using the dimensionless variables 2 u au10W [ 1 22 v , V , ] , amc12 ||Hs , u0 W0 W0 2 whereu0 is the initial amplitude of the velocity, and W0 the wavelength. The equation (2.3.6) becomes wwwvvv3 sign(mv ) 2 Pf( v ) 0 , (2.3.9) wV w] w]3 61 SOME PROPERTIES OF NONLINEAR EQUATIONS 22 Hlc02s where P 2 , and W00um|| 2 3 fv() E12 v E v E 3 v, (2.3.10a) B1 B2 WH02cs E1 Q 2 !0 , E2 Q !0 , E33 QB !0 , Q . (2.3.10b) u0 u0 ||m The equation (2.3.9) governs the motion of long transversal waves in a layer in the case of a driving force ff . Due to the cubic nonlinearity of (2.3.10a), the equation wv (2.3.9) can be reduced to modified KdV equation that contains the term v2 . w] 2 Introducing the initial condition vA(0) 0 sech V, the equation (2.3.9) admits a dilatonic behavior in some circumstances. For A0 / Acr =1, and m 1 , the initial soliton transforms into an asymmetric soliton, which becomes unstable at the finite time 8 a 8 W W ()2 1/2 , ab !0 (Figure 2.3.1). c 45 P 225 Figure 2.3.1 The dilatonic behavior of an initial soliton of amplitude A0 / Acr =1 due to a driving force fm(1 ) . The dilatonic behavior depends on the properties of the medium and on critical a2 amplitude of the initial soliton Acr . For A0 Acr the amplitude of the 15P |u0 | soliton decreases, and for A0 t Acr the amplitude of the soliton dramatically increases. When the nonlinear and dispersive effects are equilibrated, the perturbed soliton by the initial data remains unchanged and propagates without changing its identity (velocity, amplitude and shape). In Figure 2.3.2 it is represented as the explosive amplification of the dimensionless amplitude v (solution of (2.3.9)) with respect to time t [sec] and coordinate X1 [m], for three values of the ratio rA00 / Acr , namely rr = 0.98, = 1 and r = 1.2, for the case m 1. As we said, the soliton is a perfect balance between nonlinear and dispersive effects, exhibiting a remarkable survivability under conditions where a wave might normally be destroyed. INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 62 Figure 2.3.2 Amplitude of v with respect to time tX and , for r = 0.98, r = 1 and r = 1.2 ( m 1 ). 2.4 Symmetry groups of nonlinear equations The evolution equations that admit the solitonic solutions have infinitely many conservation laws. These conservation laws are related to the symmetry groups of the equations. In this section, we adopt the notations and the point of view of Bâlă. Let us consider a system of partial differential equations ()n 'Q (,xu ) 0, Q 1,2,...,l , (2.4.1) where x (,xx12 ,...,) xp , uuuu (,12 ,...,)p , and ()nn() ()n '(xu , ) (''1 ( xu , ),...,l ( xu , )) , is a differentiable function. All the derivatives of u are denoted by u()n . Any function uhx (), hD:Rop U Rq ,hhh (,12 ,...,hq ) , induces the function upr()nn ()h D 0 ()nn() called the n- th prolongation of h , which is defined by uhjj w ,,pr hD: o U ()n ()nn and for each x D ,pr h is a vector whose qp Cpn entries represent the values ofh and all its derivatives up to order n at the point x . The space DUu ()n , whose coordinates represent the independent variables, the dependent variables and the derivatives of the dependent variables up to the order n , is called the n- th order jet space of the underlying space DUu . Therefore, ' is a map from the jet space DUu ()n to R l . The system of equations (2.4.1) determine the subvariety Sxuxu {(,),(,)0()nn'() }, 63 SOME PROPERTIES OF NONLINEAR EQUATIONS of the total jet space DUu ()n . We can identify the system of equations (2.4.1) with its corresponding subvariety S . Let M uDU()n be an open set. A symmetry group of (2.4.1) is a local group of transformations G acting on M with the property that wherever uf ()x is a solution of (2.4.1) and whenever g f is defined for g G , then ug fx()is also a solution of the system. The system (2.4.1) is called invariant with respect to G . If X is a vector on M with corresponding 1-parameter group exp(HX ) that is the infinitesimal generator of the symmetry group of (2.4.1). The infinitesimal generator of the corresponding prolonged 1-parameter group pr()n [exp(H X )] ()nnd () ()n pr X|[()n prexp()H X]( x, u )| , (,xu ) dH H 0 for any (,xu()n ) M ()n , is a vector field on the n- jet space M ()n called the n- prolongation of X and denoted by pr()n X . The system (2.4.1) is called to be of maximal rank if the Jacobi matrix §·w' w' Jxu(,()n ) ¨¸QQ , , ' ¨¸xi u0 ©¹wwj of' , with respect to all the variables (,xu()n ) is of rank l whenever '(,xu()n ) 0. THEOREM 2.4.1 (Bâlă) Let pqww Xxuxu ]i (,) I (,) , ¦¦i D D i 11wwx D u be a vector field on M uDU. The n- th prolongation of X is the vector field q w pr()nn X X (, x u() ) IJ , ¦¦ D D D 1 J wu j defined on the corresponding jet space M ()nnuDU(), and J is the multi-indices J J (jj1 ,...,k ) , 1 ddjpk , 1 ddkn. The coefficient functions ID are given by pp Jn(,x uD() ) ( i uD ) i uD . IDD J I]]¦¦iJ,i ii 11 THEOREM 2.4.2 (criterion of infinitesimal invariance). Consider the system of equations (2.4.12) of maximal rank defined over M uDU. If G is a local group of transformations acting on M and ()nn() pr X[(,)]0'Q x u , Q 1,2,...,l , (2.4.2) INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 64 ()n whenever 'Q (,xu ) 0, for every infinitesimal generator X of GG , then is a symmetry group of (2.4.1). To find the symmetry group G of the system (2.4.1) we consider the vector field X on M and write the infinitesimal invariance condition (2.4.2). Then eliminate any dependence between partial derivatives of uD and write the condition (2.4.2) like polynomials in the partial derivatives of uD . Equate with zero the coefficients of the partial derivatives of uD in (2.4.2) and obtain a partial differential equations system i with respect to the unknown functions ] , ID , and this system defines the symmetry groupG of (2.4.1). Now let us study the symmetry group of the system of equations arising from the Tzitzeica equations of the surface (Bâlă) Tuu abTT u v , Tuv hT, (2.4.3) Tvv abcc TT u cc v , where the independent variables are uv and , and the dependent variable is T . We associate to (2.4.3) the integrability conditions (see section 10.2) ah hu , abahv cc , bbbv cc 0 , (2.4.4) bhcc hv , aaacc cc 0 , babhucc cc , whereaba,,cc ,... h , are invariant functions of u and v . The solutions x xuv(,), yy (,)uv, z z(,)uv of (2.4.3) and (2.4.4) define a Tzitzeica surface. Denote x1 u , x2 v andu1 T . Let DUu 2 be the second jet space attached to (2.4.3) and M uDU2 an open set. An infinitesimal generator of the symmetry group G of (2.4.3) is given by www X ] K D , (2.4.5) wwwTuv where ] ,K and D are functions of uv , and T . The theorem 2.4.1 gives the first and the second prolongations of the vector X , namely ww pr(1) X X DDuv, wTuv wT 65 SOME PROPERTIES OF NONLINEAR EQUATIONS www pr(2) X pr(1) X DDDuu uv vv , (2.4.6) wTuu wTuv wTvv where u D Duuvu() D]TKT]TKTuuv , v D Dvuvu() D]TKT]TKTvvv , uu D Duu () D]TKTu v]TKT uuu uuv , (2.4.7) uv D Duv () D]TKTu v]TKT uuv uvv , vv D Dvv ()D]TKTu v]TKT uvv vvv . The existence of a subgroup G1 of the symmetry group G , which acts on the space of the dependent variable T , and the existence of a subgroup G2 of the symmetry group G , which acts on the space of the independent variables u andv , are proved by the following theorems (Bâlă ): THEOREM 2.4.3 The Lie algebra of the infinitesimal symmetries associated to the subgroup G1 of the full symmetry group G of (2.4.3) is generated by the vector field w Y T . (2.4.8) 1 wT THEOREM 2.4.4 The general vector field of the algebra of the infinitesimal symmetries associated to the subgroup G2 of the full symmetry group G of (2.4.3) is ww Zu ]() K () v, (2.4.9) wwuv where ] and K satisfy ]aaauvuuuK ] ] 0 , ]bbbbuvuK K 2 ] u 0 , ]hhhuvK ()] uvK 0 , (2.4.10) ]aaauvccK cc cc] uK2 a cc v 0 , ]bbbuvccK cc cc K vvv K 0 . INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 66 2.5 Noether theorem In 1918, Emmy Noether proved the remarkable theorem giving a one-by-one correspondence between symmetry groups and conservation laws for the Euler– Lagrange equations (Teodorescu and Nicorovici-Porumbaru). To understand the Noether theorem, let us consider a mechanical system with n degrees of freedom, whose states are determined by the independent variable, time t , and the dependent variables (state functions or generalized coordinates) qi , in 1, 2,... , . Suppose that the equations of motion derive from the functional t1 Lq(12 , q ,..., qnn , q 12 , q ,..., q , t )d t, (2.5.1) ³t0 whereL is the Lagrangian function, qi , in 1,2,..., , the generalized velocities, and tt[,01t ] . To obtain the motion equation, we introduce the infinitesimal transformation of time ttc G t, (2.5.2) with Gt an arbitrary infinitesimal quantity. In terms of qqiic G qii, and qqiic Gq , in 1,2,..., , the variation of the functional (2.5.1) is tt11c G Lq(,,)dcccc q t t Lq (,,)d q t t, i 1,2,...,n . (2.5.3) ³³ii ii tt00c d When the Lagrangian L does not explicitly depend on time, then Gqq G. iidt Writing the Taylor series expansion, neglecting the second derivatives wwwLLL Lq(,,)iiccc q t Lq (,,) ii q t G qi G q i G t, wwwqqtii where the Einstein’s summation convention for dummy indices is used, the variation of the integral becomes t 1 ªºddwwLL§· ww LL§· G ()(Lt G qt G G q GGq qt)d t, (2.5.4) ³ «»ii¨¸¨¸ ii ddtqqwwii ww qtq ii t0 ¬¼«»©¹©¹ where in 1,2,..., . The motion equations are obtained from (2.5.4) for assuming that the variation Gt and the variation of generalized coordinates are identically zero at t t0 and t t1 . Thus, the first term in (2.5.4) vanishes, and the Lagrange equations of motion are derived wwLLd §· ¨¸ 0 . (2.5.5) wwqtqiid ©¹ 67 SOME PROPERTIES OF NONLINEAR EQUATIONS The Lagrange equations are invariant if we replace LL by D , where D is an arbitrary non-zero constant. This transformation is called the scale transformation. df The Lagrange equations are also invariant if we replace L by L , where dt ftq(,i ), is an arbitrary function. This transformation is called the gauge transformation. The transformation of independent variable under which the form of the equations of motion (2.5.5) remains invariant is a symmetry transformation. A general form of a transformation of independent variable is ttc M(), (2.5.6) and the changes of generalized coordinates are qtiicc() Qtq (,i ). (2.5.7) The functional (2.5.1) is invariant with respect to (2.5.6) if Lqqtcccc(,,)dii t c Lqqt (,,)dii t. (2.5.8) The Lagrange equations are invariant if df Lqqtcccc( , ,) Lqqt (, ccc ,). (2.5.9) ii ii dtc We can say that (2.5.6) is a symmetry transformation for a mechanical system if and only if the conditions (2.5.8) and (2.5.9) are satisfied. For the infinitesimal transformation (2.5.2) the conditions (2.5.6) and (2.5.7) yield to §·www d ¨GGGtqii qLft ¸ G(,qi ). (2.5.10) ©¹wwtqtii w d t The transformation (2.5.2) is a symmetry transformation if, for a given L , there exists a function ftq(,i ) , so that the equation (2.5.10) is satisfied. For a given L , the symmetry transformation forms a group, which represents the symmetry group of the system. Consequently, when the equations of motion (2.5.5) are satisfied, it results an equation of conservation of the form d §·wwLL ¨¸LtG qii G t G q G f 0 . (2.5.11) dtqq©¹wwii On this basis, a symmetry transformation of a mechanical system is associated with an equation of conservation. This result is proved by the Noether theorem. THEOREM 2.5.1. (Noether) If the Lagrangian of a mechanical system is invariant with respect to a continuous group of transformation with pp parameters, then exist quantities which are conserved during the evolution of the system. In the case when the Lagrangian Lqq(,) does not depend explicitly on time, it relates to the Hamiltonian H(,qp ) by INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 68 wH qi , in 1,2,..., , (2.5.12) wpi wherepi , in 1,2,..., , are the generalized momenta in the Hamilton formulation. LetCqp(, ) be an integral of motion in the Hamilton formulation. In the Lagrange formulation, this integral becomes F(,)qq Cqp (, ). The integral of motion along trajectories of the system satisfies the condition d Fqq(,) 0. (2.5.13) dt Along the trajectories system, the variation of the Lagrangian due to the transformations qqiic Gqi , i 1,2,...,n , is given by d wL GL ( Gqi ) . (2.5.14) dtqw i From (2.5.14) it follows certain conserved quantities (Teodorescu and Nicorovici- Porumbaru). Case 1. If GL 0 , the integral of motion is given by wL pqiiG . (2.5.15) wqi d Case 2. If there exists a function fq() such that GLf G()q , then the integral i dt i of motion is wwLf w f ()(Gqpii G) qi . (2.5.16) wwqqii w q i Case 3. If there exists a function g(,)qqii such that d GLgqq (,) , (2.5.17) dt ii the integral of motion becomes wL Gqgqqpqgqqiiiiii(,) G (,)i . (2.5.18) wqi We see that (2.5.15) and (2.5.16) are particular cases of the third case. THEOREM 2.5.2 (Noether) To every infinitesimal transformation of the form (2.5.10), due to a variation of the Lagrangian of the form (2.5.17), it corresponds to a conserved quantity defined by (2.5.18). 69 SOME PROPERTIES OF NONLINEAR EQUATIONS 2.6 Inverse Lagrange problem The study of the inverse problem for a given evolution equation consists of the calculus of variations to determine if this equation is identical to a Lagrange equation (Santilli 1978, 1983). Zamarreno deduced in 1992 the known Helmholtz conditions (1887) for the existence of a matrix of multipliers allowing an indirect Lagrangian representation of the Newtonian system qFqqtii (,,) . He considers the case when Fi is time independent, and gives the functional relation between the Lagrangian and a first integral of motion equations. This section presents the principal results obtained by Zamarreno. The inverse Lagrange problem consists in determining the Lagrangian from a given evolution equation. DEFINITION 2.6.1. Given a system of second order differential equations qFqqqqqqtii (12 , ,..., n , 12 , ,..., n , ) , i 1,2,...,n , (2.6.1) which describes the motion of a mechanical system with n degrees of freedom, the inverse problem consists of determining a LagrangianLq(,q, t ) such that d §·wwLL qFqqti iii(,,) ¨¸ 0, i {1,2,...,n } . (2.6.2) dtq©¹wwii q 2 We assume that the dynamic functions Fi C are defined for a time interval (ttt1 2 )] for which the generalized coordinates and velocities vary in [,qqccci and [,qqccci ] intervals of the phase space. The operator d/dt is applied over the integral trajectories of (2.6.1) and is defined as d www Fqii . (2.6.3) dtqqtwwwii Lagrange equations (2.6.2) can be written under the form wwww222LLLL Fqkk 0 , i 1,2,...,n , (2.6.4) wwqqik ww qq ik ww qt i w q i and represent an over determined system of n second order partial differential equations in Lqqt(,,) . The results are presented in the following propositions (Zamarreno). PROPOSITION 2.6.1 If the system (2.6.1) is equivalent to the Euler–Lagrange equations corresponding to a variational principle whose Lagrangian is Lqqt(,,) , then the partial derivatives of generalized momenta with respect to the generalized velocities 2 wpi w L wp j Dij Dji , (2.6.5) wwwwqqqqjiji INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 70 satisfy identically the system of equations dDij 11wwFFkk DDik jk 0 . (2.6.6) d2tqwwji 2 q PROPOSITION 2.6.2 In the same conditions as before, the equations wD()ikFF kwD()jkF k 1 §·ww§·w h wF h ¨¸qkihjh¨¸ DD 0 , (2.6.7) qq2 tqqq¨¸ wwji©¹ wwww kji©¹ are identically verified. Let us denote by Dij the multipliers that allow an indirect Lagrangian representation of (2.6.1) d §·wwLL D{ij (,,)(qqt qj F j ) ¨¸ , ij, 1,2,...,n . (2.6.8) dtq©¹wwii q Expanding the time derivatives on the right-hand side, we obtain Dij D ij . (2.6.9) In this way the equations (2.6.6) and (2.6.7) coincide according to well-known Helmholtz conditions. We also have wDij wD ik , (2.6.10) wwqqkj because 33 wDij wwLLwD ik . (2.6.11) wwwwwwwwqqqqqqqqkkijjik j Consider only regular systems for which det(Dzij ) 0 . (2.6.12) PROPOSITION 2.6.3 If D'ij is a symmetric solution of (2.6.6), then its determinant satisfies the equation d' n wF '¦ i 0 . (2.6.13) dtqi 1 wi COROLLARY 2.6.1 If Dij and Dijc are two nonsingular symmetrical matrix solutions * 1 of (2.6.5), the determinant ' |(Dij )()|Dijc is a constant of evolution of the system. 71 SOME PROPERTIES OF NONLINEAR EQUATIONS PROPOSITION 2.6.4 IfDij is a symmetric nonsingular matrix, solution of (2.6.6), then ' is a Jacobi multiplier for the dynamical system written in the equivalent form ddqqddqq dd qq dt 12 ... nn 12 ... . (2.6.14) qq12 qFF nn12 F PROPOSITION 2.6.5 IfDij is a symmetric matrix, solution of (2.6.5) and the matrix wFi F of elements is skew symmetric, the trace of any integer power of Dij are wq j constants of motion for the dynamical system. We apply the variation of constants method to integrate (2.6.6), for the uncoupled case dDwij 11wFi Fj DDij ji 0 . (2.6.15) d2tqwwji 2 q The solution of (2.6.15) is expressed as §·1 §·wFj wF D* exp¨ ¨¸ i dt ¸ . (2.6.16) ij ¨ 2 ³¨¸qq¸ ©¹©¹wwji * Here, the repeated indices do not imply summation. Writing Dij PD ij ij , we obtain * ddDPij ***ij 11wwFFkk PDij ij DPik ik DPjk jk 0 . (2.6.17) dd22ttqqwwji * From Pij P ji , and taking into account that Dij are solutions of (2.6.15), it follows * dDij 11wwFFkh** Pij DPik ik DPjh jh 0 , kjz , hiz . (2.6.18) d2tqwwji 2 q * *1 Replacing Dij from (2.6.16) and multiplication by Dij , yield dPwij 11wwFF§·§·Fj Pkkexp¨¸ dt ik ³¨¸ d2tqwwwjj¨¸ 2 qqk © ©¹¹ (2.6.19) 11wwwFFF§·§· Dhi* exp hdtjki 0,zz ,h . jh ¨¸³¨¸ 22wwwqqqii©¹©¹h Denoting by aij the expressions wwFF§·1 §·wFj at iiexp¨¸¨¸d , ij qqq¨¸2 ³¨¸ wwwjj©¹©¹i the system (2.6.19) becomes INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 72 dP 1 ij P()0aa P ,, kjh z zi. (2.6.20) d2t ik kj jh hi PROPOSITION 2.6.6. The determinant * of elements Pij , is a constant of evolution for the dynamical system. PROPOSITION 2.6.7. Sufficient conditions for the matrix (Dij ) to be diagonal are 1 aaij ji k ij const. If Fi does not depend explicitly on time, the following results hold: PROPOSITION 2.6.8 If the multipliers Dij satisfy the Helmholtz conditions and the dynamical system is autonomous, there exists a first integral I(,)qq for the dynamical system given by wI Dikq k , wqi §·ww22LL wI DikF k ¨¸ qk , (2.6.21) ©¹wwqqik ww qq ik w qi for which d0I , along the system trajectories. We can define I(,)qq as wI Dikq k , wqi qFkk§·ww F h wI DikF k ¨¸ DDih kh . (2.6.22) 2 ©¹wwqqki wqi PROPOSITION 2.6.9 The Lagrangian that allows an indirect representation for the autonomous system qFqqi iii(,), (2.6.23) by using Dij , must verify wL I qLi , (2.6.24) wqi where I(,)qq is an integral for (2.6.22). As an example, consider the simple pendulum equation g Tsin T 0 . l 73 SOME PROPERTIES OF NONLINEAR EQUATIONS The Helmholtz conditions (2.6.6) and (2.6.7) are satisfied for D ml 2 . A prime integral I is obtained by integrating (2.6.22) 1 Imlmgl 22TT (1 cos ) , 2 and the Lagrange function is obtained from (2.6.24) 1 Lmlmgl 22TT (1 cos ) . 2 Another example is given by (Santilli 1978) 1 qqqqJ(2) 2 0 , 11132 1 qqqqJ(2) 2 0 , 22232 1 qqqqqqJJJ( 22 exp( ) 2 exp( )) 0 . 3311222 This system of equations satisfies the conditions of Proposition 2.6.7, and therefore (2.6.6) admits a diagonal symmetric matrix as a solution D11 exp{ J (qq1 3 )}, D22 exp{ J (qq2 3 )}, D33 exp( Jq3 ) , which satisfy (2.6.7) The first integral is obtained from (2.6.22) cc c I qqq22exp(JJJ ) exp{ ( qqq )} 2 exp{ ( qq )} . 22331 132 2 23 The Lagrange function is determined then from (2.6.24) cc c Lq 22exp(JJJ q ) q exp{ ( qq )} q 2 exp{ ( qq )}. 22331 132 2 23 Among other works dedicated to inverse problems in mechanics we mention Chiroiu and Chiroiu (2003a), Frederiksen, Tanaka and Nakamura. 2.7 Recursion operators The existence of infinitely conserved densities for evolution equations is explained by generalized symmetries that are groups, whose infinitesimal generators depend not only on the independent and dependent variables of the system, but also the derivatives of the dependent variables. The existence of infinitely many symmetries for evolution equations of the form INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 74 uutk fuuu(,1 ,...,k1 ), (2.7.1) where uu1 x , uu2 xx , and so on, and the right-hand sides are homogeneous with respect to the scaling symmetry Txuu x O, with Ot0 , it was studied by Wang. The equations of the type (2.7.1) are called O -homogeneous equations. If an equation (2.7.1) has a nonlinear symmetry, it has infinitely many and these can be found using recursion operators. Integrable evolution equations in one space variable, like the KdV equation, admit a recursion operator, which is an operator invariant under the flow of the equation, carrying symmetries of the equation into new symmetries. In 1977, Olver provided a method for the construction of infinitely many symmetries of evolution equations, originally due to Lenard. This recursion operator maps a symmetry to a new symmetry. For the KdV equation, for example, a recursion operator defined by Im(Dx ) is given by 21 DuuD2 1 , (2.7.2) x 33xx 1 where Dx is the left inverse of Dx . Magri studied in 1978 the connections between conservation laws and symmetries from the geometric point of view. He observed that the gradients of the conserved densities are related to the theory of symmetries. This problem required the introduction to Hamiltonian operators. Magri found that some systems admit a pair of Hamiltonians. For example, the KdV equation can be written in two forms corresponding to a pair of Hamiltonians 121 uDu ()( u23 D uDu)u . txxx233x x x The Nijenhuis operator (hereditary operator) is a special kind of recursion operator found as an integrability condition. For any vector field A0 leaving the Nijenhuis j operatorA invariant, the Aj A ()A0 , j 0,1,... , leave A invariant and commute in pairs. Gelfand and Dorfman gave the relation between Hamiltonian pairs and Nijenhuis operators. For the KdV equation the operator 21 A DuuD21 , x 33xx is a Nijenhuis recursion operator and it produces the higher KdV equations, the KdV hierarchy, which shares infinitely many commuting symmetries produced by the same recursion operator j uutx A (), j 0,1,... . The recursion method can be applied also to O!0 , and Od0 . In the particular caseO 1 , we have 2 – Potential Korteweg–de Vries equation fu xxx u x , 2 – Modified Korteweg–de Vries equation fu xxx uu x , – Burgers equation fu xxx uu. 75 SOME PROPERTIES OF NONLINEAR EQUATIONS When O 2 , we have – Korteweg–de Vries equation fu xxx uu x . A selected list of integrable evolution equations, scaling symmetry transformation T , and the recursion operators are given below (Wang) – Korteweg–de Vries equation uut xxx uu x , Txuu x 2 , 21 A DuuD21 . x 33xx – Burgers equation uutxx uux , Ttxuu 2( wtx ) wu , 1 A Du, A tD tu x . 1 x x 2 xx2 – Potential Burgers equation 2 uutxx ux , Txu x , 1 A Du, A tD tu x . 1 x x 2 xx2 – Diffusion equation 2 uuutx x , Taxubu x , ab, C, 1 A uDx uxx D x . – Nonlinear diffusion equation 2 utxx D(uu / ), Taxubu x , ab, C, 1 2u A Dux D1 . uux 2 tx – Potential KdV equation INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 76 2 uut xxx3u x , Txuu x , 21 A DuDux 42xx xx . – Modified KdV equation 2 uut xxxuu x , Txuu x , 22 A DuuD22 1u . xx33x – Sine-Gordon equation uuxt sin , Taxubu x , ab, C, 22 1 A DuuDux xxxxx . – Liouville–Tzitzeica equation uuxt exp , Taxubu x , ab, C, 22 1 A DuuDux xxxxx . – Tzitzeica equation uuxt Dexp( 2 ) E exp(u ) , Txu x , 624 3 A DuuDux6( xx x ) x 9( xxx 2 uuD x xx ) x 22 42 (5uuuuuuuDxxxx 22 x xxx 13 xx 6 x xx 9 x ) x 223 (uuuuuuuuuuuDxxxxx 8 x xxxx 15 xx xxx 3 x xxx 6 x x 18 x xx ) x 3226 42020204uux xxxxx uu x xxx uu x xx u xxx uu x xx u x 1 22 2(uDx x u xxxxxx 55520uuxx xxxx u xxx uu x xxxx uuu x xx xxx 34 2 55uuuuxx x xx 2(5xxxxx uuuu xx xxx 5 x xxx 251 5).uuxxx u x D x u xx – Dispersiveless long wave system 77 SOME PROPERTIES OF NONLINEAR EQUATIONS uutx vuvx , vutx vvx , §·xux §·2u Ta ¨¸¨¸, aC , ©¹xvx ©¹v §·vuuD2 1 A xx . ¨¸1 ©¹2 vvD xx – Diffusion system 2 uutxx v, vvtx x , §·xux 2 u §·2u T ¨¸aaC¨¸, , ©¹xvx 2 v ©¹v 1 §·DvDxx A ¨¸. ©¹0 Dx – Nonlinear Schrödinger equation 22 22 vuuuvtxx r() ,,uvtxx B vuv() §·xuux T ¨¸, ©¹xvvx §·BB22vD11 u D vD v A xxx. ¨¸11 ©¹rDuDuuDvxx22 r x – Boussinesq system 18 uv , vu uu , txtxxx33x §·xuux 2 T ¨¸, ©¹xvvx 3 §·32vvDD12 2 uuD1 A xx x xx , ¨¸1 ©¹A21 3vvD xx 110 16 A DuDuDuuvD42 53 2 21 . 21 33x xxxxx3 tx Chapter 3 SOLITONS AND NONLINEAR EQUATIONS 3.1 Scope of the chapter Solitons or solitary waves are localized waves that travel without change in shape. In the mathematics literature the word soliton refers to solitary traveling waves which preserve their identities after a pair-wise collision. Solitons were first discovered in shallow water by the Scottish engineer John Scott Russell in 1844, but they exist everywhere, in many kind of systems. The study of solitons is an exciting branch in the science of nonlinear physics and their importance as nonlinear waves is well-recognized in the past two decades. The basic theory of solitons is simple and relies on mathematical methods well-known to any applied mathematician. In the last years of the nineteenth century, mathematicians as Korteweg, De Vries, Bianchi, Boussinesq, Darboux and Bäcklund have analyzed some remarkable nonlinear partial differential equations which possess the cnoidal or solitonic solutions. Engelbrecht is right when he affirms that the story of solitons is the story of three equations, namely Korteweg and de Vries, sine-Gordon and nonlinear SchrCdinger equations. In this chapter the first five evolution equations are briefly discussed and the solitonic solutions outlined. The stability and particle-like behavior of the solitons can be explained by the existence of an infinite number of conservation laws. So, these equations can be interpreted as completely integrable Hamiltonian systems in the same sense as finite dimensional integrable Hamiltonian systems, where we find for every degree of freedom a conserved quantity. The reader is referred to the remarkable monographs of Ablowitz and Segur (1981), Dodd et al. (1982), Lamb (1980), Drazin (1983), Drazin and Johnson (1989) and Engelbrecht (1991). We mention also the monographs of Carroll (1991), Nettel (1992) and Newell (1985) for enlarged topics in soliton theory. 3.2 Korteweg and de Vries equation (KdV) To explain the nature of solitons, we will consider the behavior of water waves on shallow water. The scenario could be set in one of the canals, which was the 19th century's analogue to highways nowadays. Indeed, it was in such a location that the Scottish engineer John Scott Russell first noticed a soliton in 1844. SOLITONS AND NONLINEAR EQUATIONS 79 The solitary wave, or great wave of translation, was first observed on the Edinburgh to Glasgow canal in 1834 by Russell. Russell reported his discovery to the British Association in 1844 as follows (Drazin): “I believe I shall best introduce this phenomenon by describing the circumstances of my own first acquaintance with it. I was observing the motion of a boat which was rapidly grown along a narrow channel by a pair of horses, when the boat suddenly stopped – not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished and after a chase of one or two miles I lost it in the windings of the channel. Such in the month of August 1834 was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation…”. Russell performed some experiments in the laboratory in a small-scale wave tank in order to study the phenomenon more carefully. This is shown in Figure 3.2.1, which is one of Russell's original diagrams (Dodd et al.). Figure 3.2.1 is represented as a raised area of fluid behind an obstacle. When this obstacle is removed, a long heap-shaped wave propagates down the channel. Figure 3.2.1b is identical except that the initial volume of trapped fluid is larger. In this case two solitary waves are found. If a wave is somehow initiated in such a canal, we expect that the wave rolls along the canal while it spreads out and soon ends its life as small wiggles on the surface. But, in certain conditions a soliton can be excited, and the wave will continue to roll along the canal without changing shape. It turns out that a soliton is very robust against perturbations. The bottom of the canal may be uneven and bumpy, but the soliton will gently pass all obstacles. Let us suppose the canal has depth h and note by h K (K small) the elevation of the surface above the bottom. Korteweg and de Vries derived in 1895 the partial differential equation, which governs the wave motion (Dodd et al.) 2 wK3211g w§·2 w K ¨ DKKV2 ¸ , (3.2.1) wwthx2323©¹ wx hTh3 where V , D an arbitrary constant, T the surface tensions, U the density of 3 Ug the fluid and g gravity acceleration. This equation bears the Korteweg and de Vries names, usually shortened to KdV, and becomes one of the most celebrated equations, which is related to solitons. Introducing 22DD3 g K 8, Dux [ , W t , VVl 80 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS the equation (3.2.1) becomes uuW[12uu [[[[ u 0 , (3.2.2) where the subscripts denote partial differentiation uuW w/ wW, uu[ w/ w[ , 3 3 uu[[[ w/ w[ , etc. The factor of 12 in (3.2.2) is a matter of choice and can be rescaled by the transformation u oEu J Figure 3.2.1 From Scott Russell's Report on Waves 1844 (Dodd et al.). 1 1 For E and J the equation (3.1.2) becomes 2 12 uuW[[[[6 uu 0 . (3.2.3) The equation (3.2.3) is known as the standard KdV equation. By applying a Lie transformation uk 10Kkkxkkk,, [ 32 W 4K5 , (3.2.4) we may write the KdV equation in the forms uuuuW[[[[r60 , uuuuW r(1 )[ r [[[ 0 , (3.2.5) by choosing in an adequate way the constants kii , 0,1,2,3,4,5. Returning to the equation (3.2.3) we look for a solution with the permanent uu (),TT [c W, (3.2.6) where c is the velocity of wave. Substitution of (3.2.6) into (3.2.5) gives cuc60uu c u ccc , SOLITONS AND NONLINEAR EQUATIONS 81 where prime means the differentiation with respect to the new variable T . The above equation becomes cuuuc3(2 ) c ccc 0 . Integration with respect to T yields uucucc 3 2 A , where A is an integration constant. Multiplying this equation with uc and integrating with respect to T we have 11 uuc23 cuAu 2 B , (3.2.7) 22 with B an integration constant. The equation (3.2.7) can be reduced, in certain 32 conditions, to the Weierstrass equation (1.4.8). For gg23!27 0 , the solution is expressed in terms of the elliptic Jacobi function. At the limit m 1 , the solution of the KdV equation is described in terms of hyperbolic function 2 ()z = eee313()sech(*, um) . The boundary conditions uu,,ccc u o 0 as T of, give AB 0 , and 1 1 uuucc22 ( ) . (3.2.8) 22 From duu cdT we have dduu T , ³³uc uuc2 or 21 T auctanh( 2 ) T0 , cc with T0 an integration constant. 1 Substituting uc sech 2 I into the above equation we have 2 21 2 T actanh( (1 sech I )) T0 , cc or 212 T actanh( tanhI ) T0 , cc 1 with I c(TT ) . Therefore, we obtain the solution 2 0 82 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 11 uc(T ) sech2 { c (TT )} , (3.2.9) 22 0 for any c t 0 and T0 . These constants are determined from the initial conditions uuu(0) 00,(c 0) c . We see from (3.2.9) that the amplitude of the wave depends on the velocity. Thus, the wave having higher amplitude travels faster than the wave having lower amplitude. Similarly the solution of (3.2.2) was found looking at solutions of the form uu ()T, with T at [Z G and G the phase constant. Requiring that u and its derivatives to be zero as ||Tof the equation (3.2.2) may be integrated to give the solution 11 ua 22sech ª aaa[WG3 º . (3.2.10) 42¬ ¼ The constants a and GG are arbitrary, playing the role of a phase. The solution (3.2.10) represents a solitary wave, which has the shape as Russell observed. This wave propagates at a constant velocity without change of shape. Its velocity is Z/ a 1 a2 , and depends on amplitude and vice versa the amplitude depends on the velocity. Figure 3.2.2 represents the solution (3.2.10) for three values of a. For smaller values of a , the wave is low and broad, and becomes narrow and sharper asa increases. a = 1 aa = 0.37 = 0.15 ( G = 0.5) Figure 3.2.2 Three solitary waves (3.2.10) for different values of a. Up until this point we have talked about solitary wave solutions, which propagate without change of form and have some localized shape. But there are many equations, which have solitary wave solutions. The word soliton first appears in the paper of Zabusky and Kruskal. They have studied the problem of Fermi, Pasta and Ulam (1955) of motion of a line of identical particles of unit mass, with fixed end points, and put into evidence a remarkable property of the solitary solutions (Kawahara and Takaoka). Their interaction with one another as they passed through the cycles of evolution forced by the periodic boundary conditions shows no change in the form, but only a small change in the phase. Zabusky and Kruskal called these solitary waves solitons where the ending ‘on’ is Greek for particle. This particle-like behavior does not depend on periodic boundary conditions. Zabusky and Kruskal numerically integrated the KdV equation on boundary conditions u o[of0 as and considered two well-separated solitary wave SOLITONS AND NONLINEAR EQUATIONS 83 solutions of the KdV equation as initial data with different velocities vv1 ! 2 . The taller and therefore faster wave v1 is situated on the left. As time increases throughout the numerical integration, the tall, fast soliton overtakes the broad, slow one, as shown in Figure 3.2.3. A larger soliton travels with a high velocity so that it overtakes the smaller one and after the collision they reappear without changing their form identity. Figure 3.2.3 The interaction of two solitons running in the same direction. The collision does not result in any small waves or wiggles after the interaction. The two solitons look exactly as they did before the collision. This particle-like behavior is generic for the soliton (Munteanu 2003). Therefore a phenomenological definition of the soliton could be a localized wave that does not radiate (generate small waves) during collisions. The water wave soliton is a result of a dynamic balance between dispersion, namely the wave's tendency to spread out, and nonlinear effects. In order to substantiate this statement, we have to pass to some mathematics. 84 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS The dynamics of water waves in shallow water is described mathematically by the Korteveg–de Vries (KdV) equation (3.2.3). The second and the third term in the equation are the nonlinear and the dispersive term, respectively. Let us first investigate the effect of the dispersive term. Thus, we neglect the nonlinear term in the KdV equation. This leaves us with the following: uuW[[[ 0 . We know that waves with different wavelengths travel with different velocities. Since the initial wave is composed of many small waves with different wavelengths, it will soon spread out in the many components and can no longer be described as an entity or object. Now let us see the effect of the nonlinear term. We neglect the dispersive term in the KdV equation, which leaves us with the following uuuW[60 . The top of the wave moves faster than the low sides and this causes the wave to shock in the same way as the waves we see on the beach. This behavior is known as breaking the waves. When both the dispersive and the nonlinear term are present in the equation the two effects can neutralize each other. If the water wave has a special shape the effects are exactly counterbalanced and the wave rolls along undistorted. For the KdV equation, there are four such linear independent symmetries, namely arbitrary translations in x and t , Galilean gauge and scaling. A conservation law for the KdV equation (3.2.3) is (Kruskal, Zabusky and Kruskal) DUW[ D F 0 , (3.2.11) whereU is called the conserved density and F is called the conserved flux. The expressions for the conservation of momentum and energy are known as u2 Du D() u 0 , W[x [2 u23u2 u DDuu() ([ ). (3.2.12) W[[22x 3 Zabusky and Kruskal continued searching and found two more densities or order 2 and 3 expressed in terms of highest derivative. They missed a conserved density of order 4. Miura, Gardner and Kruskal found a conserved density of order 5, and filled in the missing order 4. After the order 6 and 7 were found, Miura was fairly certain that there was an infinite number. Later it was proved that the conjecture if an equation has one nontrivial symmetry, it has infinitely many, is true under certain technical conditions (Wang). Miura found that the modified Korteweg–de Vries equation (MKdV) 2 vvW [[[ vv [ , (3.2.13) has an infinite number of conserved densities. He introduced the transformation SOLITONS AND NONLINEAR EQUATIONS 85 2 uv 6 v[ , (3.2.14) which now bears his name. Applying the Miura transformation to (3.2.14) we have (Wang) 2 uuW()(26D(( [[[ uuv [ [ vv W [[[ vv [ )) , (3.2.15) from which we see that, if v(,)[ W is a solution of (3.2.13), then u(,)[ W is a solution of (3.2.3). From this observation, the famous inverse scattering method was developed and the Lax pair was found (Lax, Lax and Phillips). Gardner also observed that the KdV equation might be written in a Hamiltonian u 2 way. Notice that the Hamiltonian of KdV is H . 2 The integrability of the KdV equation can be understood in the same sense as finite dimensional integrable Hamiltonian systems, where we can find for every degree of freedom a conserved quantity named the action. Let us return to the linearised KdV equation uuutxxxx 0 , (3.2.16) 3 whereZ kk . The term uxxx contains the dispersion of waves. Let us ignore the dispersive term in the KdV equation (3.2.5) uuutx(1)0 , (3.2.17) and introduce the initial conditions ux(,0) f () x. (3.2.18) To solve this problem, consider the equation uuutx0 0 , with the solution uuxut ( 0 ) , which propagates with the velocity u0 . For ux(,0) fx(), the complete solution is ux(,)tfxut (0 ). This solution leads to the idea to consider a functional equation of the form (Dodd et al.) ufxu [(1) t] . (3.2.19) From uutfxx (1 ) c , (3.2.20) utuutt [1 ] f c , (3.2.21) we obtain the equation to be verified by the solution of (3.2.17) [(1)](1)uuutx tfc 0 . (3.2.22) The equation (3.2.19) is an alternative form of (3.2.17), and can be solved by the characteristics method. Dodd and his workers, Eilbeck, Gibbon and Morris, have shown that (3.2.19) is solvable for triangle initial conditions of the form 86 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS ux0 ,01 x , ° fx() ® u0 (2 x ), 1 x 2, (3.2.23) ° ¯ 0,xx ! 0; 2. From (3.2.19) we have uut0 (),0K dK utd 1, ° ut(,)K ® u0 (2 K ut ),1 dKd ut 2, (3.2.24) ° ¯ 0, in rest, where x t K. The solutions are u K u 0 , 0 ,0dKut d 1, ° 1 ut °1 ut ° 0 ° 0 °u0 (2K ) ° u0 u ® , uK ® ,1 Kdut 2, (3.2.25) ° 1 ut0 °1 ut0 ° 0, ° 0, in rest. ° ° ¯° ¯° The behavior of the solution (3.2.25) may be understood intuitively from (3.2.19). Consider a wave traveling with the velocity (u +1). The apex of the triangle overtakes the lower points if u increases. The wave becomes multi-valued after the breaking time 1 t , when the wave breaks. The conclusion is that the absence of the dispersive u0 term yields to a discontinuous (shock) behavior of waves. 2 By inclusion into the equation of the term G uxxx , even for small values of G , the third derivative is very large and the shocks never form (Zabusky and Kruskal). Inclusion into the equation of a dissipative term Guxx instead of the term uxxx , leads to the Burgers equation which is a nonlinear diffusion equation uuuutx Gxx . (3.2.26) 3.3 Derivation of the KdV equation Consider the irrotational bidimensional motion of waves on the surface of an ideal and incompressible fluid, generated by a disturbance. The fluid occupies a volume V at time t , and a hard horizontal bed bounds it below. The upper boundary is a free surface (Lamb, Miura). Consider an orthogonal system of coordinates xOz , with the vertical axis Oz , and the axis Ox oriented along the fixed bed. The components of velocity are G vxzt(,,) ( vxz (,,),(,,)) xzt v xzt . The equations of the hard bed and of the free surface are SOLITONS AND NONLINEAR EQUATIONS 87 Sz1 :0 , Sfxztzh2 :(,,) K (,)0 xt , whereh is the depth of the fluid, and K is an unknown function that has to be determined. The volume occupied by the fluid is Vxz {( , )uK R [0, hxt ( , )]} . The velocity of each particle is derived from a potential IufoI:[0,)R,CV 2 , G so that vxzt(,,) grad(,,).I xzt The continuity equation is IIxx zz 0 , (,,)xztu V [0,) f. (3.3.1) The Lagrange equation on a characteristic curve is written as 1 p I()()co I22 I gz h nst. txz2 U An atmospheric pressure p0 0 is acting on the free surface. So, the boundary condition on S2 is written as 1 I()() I22 I gz h 0 , (3.3.2) txz2 forzh K(,) xt . Another boundary condition on S2 is derived from the Euler– Lagrange condition, according with, for a material surface, we have f 0 KIKItxxz 0 . (3.3.3) We add the condition of a fixed surface S1 , vxz (,0,)t 0, that is Iz 0 , (3.3.4) forz 0 . Relations (3.3.1)–(3.3.4) are enough to determine the partial differential equation that must be verified by Kufo:R [0, ) R . So, we introduce the dimensionless quantities x lx00,, z hz vx ghv 0xz , v ghv0 z, la I(,,)xzt ghI (,,), xzt h 0 l K(,)xt aK00 (,), xt t t , gh wherel is the length of the disturbance, h the depth of the fluid, a the amplitude of the disturbance, g the gravitational force constant, and gh the velocity. Noting with a superposed bar the quantities that depend on x00, z and t0 , the equations (3.3.1)–(3.3.4) become 88 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS GI2 I 0 , 0,xx00 0, zz 00 1 KI DGI()2 22 I 0, 00,tzx002 0,0,0 2 (), I0,ztxx000 GKDIK0, 0, 0, 0 for zx000 1(,DK t0 ) , and I 0, 0,z0 ah forz 0 , with D , G . 0 hl Dropping the bars, the motion equations and the boundary conditions are 2 GIxx I zz 0, 1 2 22 ½ KItzx D()0, G I I ° 2 ¾ forz 1DK , (3.3.5) 2 ° Iztxx GKDIK(),¿ Iz 0, forz 0. For small amplitudes ( Do00), and for large wavelengths ( Go ) we suppose that G2 O() D. For arbitrary DG, , the new variables are introduced DDD [ (),x tt W , (3.3.6) GG that yield a reference system that moves with a unit velocity at z 0 . For G O( D) , we have W O(1) for large moments of time. By changing the function D I[W(,,)zxzt I (,,), K[W (,) K (,),xt (3.3.7) G the equations (3.3.5) become IDIzz [[ 0, 1 22 K DI I ()0,for1 Iz DI z DK, W[2 [ (3.3.8) Iz DKDKDIK(),for[W[[ z 1,DK Iz 0, forz 0. Looking for the solutions of the form f (,,)z n (,,)z , (3.3.9) I[W ¦ DIn [W n 0 it follows that, for fixed [W, , and z DK[0,1 ] SOLITONS AND NONLINEAR EQUATIONS 89 z2 I[W(,,)z T (,) [W DT [ (,) [W T (,)] [W 012 o ,[[ (3.3.10) zz24 zO23[(,)T [W T (,) [W T ] (), D 21224,[[ 0, [[[ whereTTT120,, are arbitrary functions that depend onz . The conditions (3.3.9), 2 3 (3.3.10) on the surface z 1DKDK1 O( D) lead, further, to the partial differential equations verified by KK01, ... . Note that the equation verified by K0 is the KdV equation written under the form 1 KKKK32 0 . (3.3.11) 3 0,[[[ 0 0,[W 0, We observe that (3.3.11) may be written under the form pq q K KK K 0 , (3.3.12) W rr[3 [[[ 1 where pq 9, , r 1 . 6 Equation (3.3.12) is equivalent to the equation pq q K KK K 0 , (3.3.13) txxrr3 xx for a change of variable qr p W txxt,,(,)( [ K K[,) W . qr p Thus, equation (3.3.11) can be reduced to the standard form of the KdV equation uuuutxxxx6 0 , (3.3.14) by the change of variables 2 [ x,6,(,)(, W tu K [W xt) . 0 3 Generally, by applying a Lie transformation ukuxtk(,)[W 01 (,) , [ kx23 k,, W kt 4 k5 and choosing the constants k0 ,...,k5 , in a convenient way, the equation (3.3.14) becomes uuuuW[[[[r60 , or 90 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS uuuuW r(1 )[ r [[[ 0 . 3.4 Scattering problem for the KdV equation Consider the KdV equation uuuutxxxx60 . A solution of the KdV equation for imposed conditions uu,,cccu o 0, as x of, is 2 ux( ,0) f ( x ) U0 sech x, (3.4.1) at the initial moment of time t 0 . This is the reason for choosing (3.4.1) as the potential function in the Schrödinger equation (1.2.13). Thus, we have 22 M'' [kU 0 sech ( x )] M 0 . (3.4.2) To solve a direct scattering problem means to find the fundamental solutions (1.2.14), the coefficients cij (1.2.16), and the transmission and reflection coefficients (1.2.19) and (1.2.20). Using the change of function ik M()x Cxyk sech()()x , (3.4.3) the equation (3.4.2) becomes 22 yk''2i tanh()'[ xykkUxy i0 sech()] 0 . (3.4.4) The change of variable 1 tanh(x ) z (1,1),x R, (3.4.5) 2 transforms (3.4.4) into 2 zz(1)''[(1i)2(1i)]'( y k kzy k i k U0 ) y 0 , where yz() yx (). This equation is written as zz(1)''[(1JDEDE y )]'zy y 0, (3.4.6) where J 1i,k 1 1 D ikU , (3.4.7) 240 11 E i,kU 240 and admits as solution the hypergeometric series SOLITONS AND NONLINEAR EQUATIONS 91 DE yz( ) F (DEJ , , ; z ) 1 z ... J DD( 1)...( Dnn 1) EE ( 1)...( E 1) ...z n ... . (3.4.8) JJ(1)...( Jn 1) 1 This series is convergent for z 1 . We have also U t . From mentioned 0 4 changes (3.4.3) and (3.4.5), the solution of (3.4.2) becomes exp(x ) exp(xx ) 1 tanh( ) M(;)xk C ( )(,,;)ik FDEJ , (3.4.9) k 22 where the complex constants are given by (3.4.7), and Ck is an arbitrary constant. The hypergeometric series has the following property (Abramowitz and Stegun) *J*JDE() ( ) Fz(,,;)DEJ F(,,DEDEJ 1;1) z *JD*JE()() (3.4.10) *J*JDE() ( ) (1zF ) JDE (JDJEJDE , , 1;1z ), *D*E()() for any z with arg 1 Sz , and * is the Euler function. Thus, the solution (3.4.9) becomes exp(xx ) exp( ) *J*JDE ( ) ( ) M(;)x kC ( )ik F k 2()(*JD*JE) 1 (3.4.11) exp(x )*J*JDE ( ) ( ) CF()ik , k 2()()*D*E 2 where 1tanh() x FF (,,DEDEJ 1; ), 1 2 1tanh() x FF (,, JD JE JDE 1; ) . 2 2 Taking x of into (3.4.9), and x of into (3.4.11), we write exp(ikx ) M#(;xk ) C,, x of k 2ik exp(*J*JDE ikx ) ( ) ( ) M#(;)xk C (3.4.12) k 2ik *JD*JE()() exp(ikx )*J*JDE ( ) ( ) oCx,.f k 2ik *D*E()() 92 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS To determine the fundamental functions fxkii (, ), 1,2,we observe that M(;x k )may be written as a linear superposition of them. It follows that there exist the functions A(),()kBkso that M(;)x kAkfxkBkfxk ()12 (; ) () (; ). According to (1.2.14), the properties of M at infinity are given by M#(x ; k ) A ( k )exp(i kx ) B ( k )[ c11 ( k )exp(i kx ) c12 ( k )exp( i kx )], forx of , (3.4.13) M#(x ;kAkckkxckkxBkkx ) ( )[21 ( )exp(i )22 ( )exp( i )] ( )exp( i ), forx of . Comparing (3.4.12)1 to (3.4.13) 1 it results Bkc()12 () k 0, ik Ak() Bkc ()11 () k Ck 2 , and *(1 ikk ) * ( i ) ck12() ck 21 () , (3.4.14) 1111 *(ikU * )(i kU ) 242004 ik Bk() 0,() Ak Ck 2. Consequently, we have M(;)x kx1 tanh( ) fxk( , ) [exp(x )DEJ exp( x )]ik F ( , , ; ) . (3.4.15) 1 Ak() 2 Comparing (3.4.12)22 to (3.4.13) , and using (3.4.14) we have C *J*JDE() ( ) Akc() () k Bk () k , 22 2(*JD*JE )( ) C *J*JDE() ( ) Akc() () k k . 21 2()()*D*E Therefore, SOLITONS AND NONLINEAR EQUATIONS 93 *(1 ikk ) * ( i ) ck12() ck 21 () , 1111 *(ikU * )(i kU ) 242004 *(1 ikk ) * (i ) ck22 () , (3.4.16) 1111 *()()UU * 242400 *(1 ikk ) * ( i ) ck11 () . 1111 *()()UU * 242400 Using (1.2.17) and (1.2.18) and the property of Euler function, the transmission and reflection coefficients are given by 1111 *(ikU * )(i kU ) 242004 Tk() , *(1 ikk ) * ( i ) *(1 ik ) 1 1 1 1 1 Rk() *( ik U )( * i k U )cos( S U ), R S*(1 ik ) 2 004 2 4 0 4 *(ik ) 1 1 1 1 1 Rk() * ( ik U )( * i k U )cos( S U ). (3.4.17) L S*(i) k 200 4 2 4 0 4 The fundamental solution f2 comes from (1.2.16a), wheref1 is given by (3.4.15), andcc11, 12 , by (3.4.16). Looking at the expression (3.4.17) we see that certain values of U0 yield to vanished reflection coefficients UN0 (1),NN 1. (3.4.18) In this case, the transmission coefficient is *(ikN 1)(i * kN ) (i kN )(i kN 1)...(i) k Tk() , (3.4.19) *(1i)(i)kk * (i)(i kk 1)...(i kN ) and the equation (3.4.2) is written as M'' [kNN22 ( 1)sech x ] M 0 . (3.4.20) The bounded solutions of this equation are the associated Legendre functions, knn i , 1,..., N , n dn M(x ,inPT ) n ( ) (1 T2 )2 PTn ( ), 1,..., N, (3.4.21) NNdT n where 1dN PT() (T2 1)N , N NT!2NN d 94 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS are the Legendre polynomial of degree , and TN tanh x . Consider now the poles of transmission coefficient (3.4.17). Since the Euler function admits negative integers as poles, we can say that, if k is pole for Tk( ) then there exists m N such that 11 ikU m. 240 Let us choose k so that k R , or k i,N N! 0. In this way we find a finite number of poles 11 kUmm iN , N ( ), 0,...,M 1, (3.4.22) mm m 0 42 11 11 ° UU00,for N, ° 42 42 M ® ªº11 11 ° UU 1, for N. °«»00 ¯¬¼42 42 In particular, for (3.4.18) the poles of the transmission coefficient are knnn i , 1,..., N , and solutions in these points are given by (3.4.21). In the caseN 1 , the equation (3.4.2) becomes M'' [kx22 2sech ] M 0 . The k i coefficients of reflection vanish, and the transmission coefficient is Tk() , k i accordingly to (3.4.19), and has a single pole k1 i . The solution is d M()x PT12 () 1 T PT (), 11dT 1d PT() ( T2 1),Tx tanh , 1 2dT or M()x sech()x . In the case N 2 , the equation (3.4.2) becomes M'' [kx22 6 sec h ] M 0 . The coefficients of reflection vanish, and the transmission coefficient is (2 ikk )(1 i ) Tk() , (2 ikk )(i 1) with poles k1 i ,k2 2i . In this case the solutions are 1 M12(x ) Px (tanh ) 3tanh x sechx . SOLITONS AND NONLINEAR EQUATIONS 95 3.5 Inverse scattering problem for the KdV equation Let us return to the relationship between the solution of the KdV equation uuuutxxxx6 0 , (3.5.1) with imposed conditions uu,0x o at x of, and the initial condition as an N - soliton profile, N N* ux( ,0) f ( x ) NN ( 1)sech2 ( x ) , (3.5.2) and the solution of the Schrödinger equation 2 MMxx (,)xt [ k uxt (,)](,) xt 0. (3.5.3) This relationship is expressed by (1.2.10) Mtx M463xxx uu Mx M . (3.5.4) The fundamental solutions fii ,1, 2 , satisfy the conditions fxkt1 ( , ; )|of exp(i kx ), at x , fxkt2 (,;)| exp(i kx ),at x of, (3.5.5) and yield f21(, xkt ;) c111 (;) kt f (, xkt ;) c21 (;) kt f (, x kt ;), f12(, xkt ;) c122 (;) kt f (, x kt ;) c22 (;) kt f (, xkt ;), (3.5.6) where cij are related to the reflection and the transmission coefficients by ckt11 (;) RktR (;) , ckt12 (;) ckt22 (;) RktL (;) , (3.5.7) ckt21 (;) 1 Tk(;)t . (3.5.8) ckt12 (;) The fundamental solutions are typically defined by f f( x , k ; t ) exp(i kx ) A ( x , x '; t )exp(i kx ')d x ' , 1 ³ R x x f( x , k ; t ) exp( i kx ) A ( x , x '; t ) exp( i kx ')d x ', (3.5.9) 2 ³ L f where the kernels AR , AL verify the equations (1.3.16a) and (1.3.16b) 96 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS f rx( yt ;) rx ( '' ytAxxtx ;) (, '';)d'' Axyt (, ;) 0, x y, RRRR³ x x rx(;)('';)(,'';)d''(,;)0, yt rx ytAxxtx Axyt x ! y, (3.5.10) LLLL³ f where 1 f rzt( ; ) Rkt ( ; )exp(i kzk )d , RR³ 2S f 1 f rzt(;) Rkt (;)exp(i kzk )d , LL³ 2S f when Tk() admits poles in the upper half-plane. Analogously, it results f :(;)('';)(,'';)d''(,;)xyt : x ytAxxtx Axyt 0 , x y , (3.5.11a) RRRR³ x x :(xyt ; ) : ( x '' ytAxxtx ; ) ( , ''; )d '' Axyt ( , ; ) 0 , x ! y . (3.5.11b) LLLL³ f In the above relations we have 1 f n :(zt ; ) R ( kt , )exp(i kz )d k i m (i N ; t )exp(N z ) , RR³ ¦ Rlll 2S f l 1 1 f n :(zt ; ) R ( kt , )exp( i kz )d k i m (i N ; t )exp(N z ) , (3.5.12) LL³ ¦ Llll 2S f l 1 ckt(;) f mkt(;) i11 l {[(;;)]d}fxkt21 x , Rl l ³ 1 l ckt12 (;)l f ckt(;) f mkt(;) i22 l {[fxkt (;;)]d}2 x1 , (3.5.13) Ll l ³ 2 l ckt12 (;)l f whenTk() admit poles klnll iN , 1,..., . Consider now the relation between the potential function ux(,)t and the kernels AR and AL ddAA uxt(,) 2RL (,) xt 2 (,) xt , (3.5.14) ddxx and SOLITONS AND NONLINEAR EQUATIONS 97 M(,,)xkt hkt (,) f (,,) xkt 1 (3.5.15) hkt(,)[(;) c21 kt f 2 (, x kt ;) c 22 (;) kt f 2 (,;)]. xkt Due to the fact thatf1 is a solution of (3.5.3), namely an eigenfunction of L given by (1.2.7), associated to the eigenvalue k 2 which does not depend on time t, we may say that M is also an eigenfunction for the same eigenvalue, and then M is a solution for (3.5.4) Mtx M463xxx uu Mx M. From the conditions uu,0x o at x of, we have lim (Mtxxx 4 M ) 0 . (3.5.16) x of Substitution (3.5.15) into (3.5.16) leads to 3 lim[(hhkkxt 4 i )exp(i )] 0 , xof 3 and thus, hhkt 4i , or hkt(,) hk (,0)exp(4i kt3 ). (3.5.17) We can write on the basis of (3.5.17) 3 limh [ c21,t exp(i kx ) 8i k c22 )exp( i kx )] 0 , xof and obtain 3 cc21,tt 0,22, 8i kc22 . Therefore cktck21 (,) 21 (,0), 3 cktck22 (,) 22 (,0)exp(8i kt ), (3.5.18) 3 cktck11 (,) 11 (,0)exp(8i kt ). According to (3.5.13) we have 3 mktmkRl( l , ) Rl ( ,0)exp(8i kt ) , 3 mktLl(,) l mkLl (,0)exp(8i) kt. In a similar way, it results 3 RktRl(,) Rk R (,0)exp(8i) kt, 3 Rktll( , ) R L ( k ,0)exp( 8i kt ) , and 98 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 1 f n :(zt ; ) R ( k ;0)exp[i( kz 8 kt3 )]d k m (i N ;0)exp(8 NN3t z ) , RR³ ¦ Rllll 2S f l 1 1 f n :(zt ; ) R ( k ,0)exp[ i( kz 8 kt3 )]d k m (i N ;0)exp( NN 83t z ) . (3.5.19) LL³ ¦ Lllll 2S f l 1 In regards to the form of (3.5.19) for the initial conditions (3.5.2), the equation (3.5.3) at t 0 is written as 2 Mxx (xkuxx ,0) ( ( ,0))M ( ,0) 0 . (3.5.20) In the section 2.2 we found the values RkRL(,0) Rk (,0)0 , (3.5.21) with poles klll i , 1,..., N. The corresponding solution is the generalized Legendre function (3.4.21) l M(,i)x lP N (tanh) x. (3.5.22) The functions (3.5.19) become N (;)zt m (i;0)exp(8 l lt3 lz ), :RRl ¦ l 1 N 3 :LLl(zt ; ) ¦ m (i l ;0)exp( 8 lt lz ) , (3.5.23) l 1 and (3.5.11) take the form of Marchenko equations NNf XRN(,) xtZ () y X R (,) xtZcc N ()(,;)d yAxxtxcc cc Axyt (,;)0, x y , ¦¦nn³ n nR R nn 11x NNx XxtZyLL(,) () XxtZyAxxtx L (cc ,) L () (,cc ;)d cc Axyt (, ;) 0, x ! y . (3.5.24) ¦¦nn³ n nL L nn 11f Here we have R 3 X nR(xt , ) mn (i n ,0)exp(8 nt nx ) , R Zn ()yn exp(y ), L 3 X nL(xt , ) mn (i n ,0)exp( 8 nt nx ) , L Zn ()y exp()ny , (3.5.25) where the summation index is n. If we write the solution of (3.5.24) as SOLITONS AND NONLINEAR EQUATIONS 99 N A (,xyt ,) LRR (,) xtZ () y, (3.5.26) Rn ¦ n n 1 the equation (3.5.24) 1 reduces to an algebraic system in unknowns R Lxtn ( , ), n 1,..., N, of the form N f X RR(,)xt L (,) xt (,) xt Z RR ( xcc ) X ( x cc ,)d t x cc 0, n 1,..., N. nn¦ ³ mn m 1 x With (3.5.25), it is easy to discover that this system can be written under the form AxtL(,)R (,) xt Bxt (,) 0, (3.5.27) where R A(,)xt MN , Bxt(,) MN ,1 ,,Lxt(,) MN ,1 1 A (xt , ) G m (i n ,0) exp[8mt3 ( m nx ) ], mn , 1,..., N, mn mn Rn mn 3 BxtnR( , ) mn (i n ,0)exp(8 ntnxn ), 1,..., N. (3.5.28) The solution of (3.5.28) is N LxtR (,) A1 (,) xtBxtm (,), 1,..., N. mm ¦ nn n 1 Return to (3.5.26) and have N A (,xyt ,) LRT (,)exp( xt ny ) E () yLR (,) xt , Rn ¦ n 1 where Ey() MN ,1 Eyn ( ) exp( ny ), nN 1,..., . Thus TR AxxtR (,,) E () xLxt (,) NN TT1 ¦¦ExLxtmm( ) ( , ) Ex m ( )[ A mnn ( xtBxt , )] ( , ), mmn 1,1 if we take into account that d A (xt , ) m (i n ,0)exp[8 mt3 ( m n )] B ( xtE , )T ( x ), dx mn Rn m n 100 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS N dd A (,)xt A11 (,) xt A (,) xt tr[ A (,) xt Axt (,)] Rm¦ nnddx m x mn,1 (3.5.29) 1d d [det(,)]Axt [lndet(,)]. Axt detAxt ( , ) d x dx The solution of (3.5.24) 2 is found in a similar way. The potential function u is according to (3.5.14) d2 uxt(,) 2 lndet(,)Axt , (3.5.30) dx with the matrix A defined by (3.5.28)1 . Return now to the equation (3.5.1) and consider N 1 , and uu,0x o , as x of and the initial condition ux( ,0) 2sech2 ( x ) . The Schrödinger equation (3.5.20) is given by 2 Mxx (xk ,0) ( 2sech xx )M ( ,0) 0 . From (3.5.21) we have (1 ik ) Rk(,0) Rk (,0)0 , Tk(,0) , RL (ik 1) and M()x sech()x . (3.5.31) Also, the functions (3.5.23) become :RR(zt ; ) ml (i;0)exp(8 t z ) , :LL(zt ; ) ml (i;0)exp( 8t z ) . To calculate the normalized constants at t 0 , we need the fundamental solutions fxj (,i;0), j 1,2,of (3.5.1) that have the properties fx1 (,i,0)exp()# x as x of, fx2 ( ,i,0)#of exp( x ) as x . (3.5.32) Because (3.5.32) are solutions for (3.5.1) it results fxjj( ,i;0) c sech( x ), j 1,2 . 1 From (3.5.3) we obtain cc . So, we can write 12 2 1 fx( ,i;0) fx ( ,i;0) sech(x ) , (3.5.33) 122 and, accordingly to (3.5.13) f 1 mm(i,0) (i,0) { [ sech21 ( xx )]d } 2 . RL11³ f 4 Equations (3.5.24) become SOLITONS AND NONLINEAR EQUATIONS 101 f 2exp(8txy ) 2exp(8 txyAxxtxAxyt cc ) ( , cc ; )d cc ( , ; ) 0, xy , ³ RR x x 2exp( 8txy ) 2exp(8 tx cc yAxxtx ) ( , cc ; )d cc Axyt ( , ; ) 0, x ! y . ³ LL f To determine the solution u we solve one of the above equations and obtain exp(8txy ) Axyt(, ,) 2 , R 1exp(82)tx and them from (3.5.30) ux( ,txt ) 2sech( 4 ) . (3.5.34) 3.6 Multi-soliton solutions of the KdV equation Two-soliton solutions for (3.5.1) are derived for N 2 , and uu,0x o at x of, and the initial condition ux(,0) 6sech()2 x , (Data and Tanaka). The Schrödinger equation is given by (3.5.20). From (3.5.21) we have RkRL(,0) Rk (,0)0 , (1i)(2i)kk Tk(,0) . (ikk 1)(i 2) The fundamental solutions fxjj( ,i;0), fx ( ,2i;0), j 1,2 , of (3.5.1) have the properties fxj ( ,i,0)#orf exp(B x ) for x , j 1, 2 , (3.6.1a) fxj ( ,2i,0)#o exp(B 2x ) for x rf, j 1, 2 , (3.6.1b) and then we have i fxjj(,i;0) cM (,i;0), x j 1,2, 2i fxjj( ,2i;0) cM ( x ,2i;0), j 1,2. From (3.6.1) it results c j 1 1 ccii , cc22ii . 126 1212 Therefore, 102 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 1 fxj (,i;0) tanh()sech(),x x 2 (3.6.2) 1 fx(,2i;0) sech().2 x j 4 From (3.5.13) it results f 1 mm(i,0) (i,0) { [ tanh22 ( xxx )sech ( )]d } 1 6 , RL11³ f 4 f 1 mm(2i,0) (2i,0) { [ sech41 (xx )]d } 12, RL22³ f 16 and (3.5.23) become :R (zt ; ) 6exp(8 t z ) 12exp(64 t 2 z ), :L (zt ; ) 6exp( 8 t z ) 12exp( 64t 2 z ). Next we solve the first Marchenko equation (3.5.24) 6exp(8txy ) 12exp(64 t 2 x 2 z ) AxytR ( , ; ) u f (3.6.3) u[2exp(8tx '' y ) 12exp(64 t 2 x '' yAxxtx )] ( , ''; )d '' 0, ³ R x for x y . Assuming the solutions of the form (3.5.26) RR AR (,xyt ,) L12 (,)exp( xt y ) L (,)exp(2) xt y , the equation (3.6.3) is reduced to (3.5.26), where ªº1 3exp(8tx 2 ) 2exp(8 tx 3 ) Axt(,) «», ¬¼4exp(64tx 3 ) 1 3exp(64 tx 4 ) ªºLxtR (,) ª 6exp(8tx ) º LxtR (,) 1 , Bxt(,) . «»R « » ¬¼Lxt2 (,) ¬12exp(64tx 2 )¼ The solution of (3.5.26) is given by R 6 ª exp(72tx 5 ) exp(8 tx ) º Lxt(,) « » , detAxt ( , ) ¬2[exp(64tx 2 ) exp(72 tx 4 )]¼ where detA (xt , ) 1 3exp(8 tx 2 ) 3exp(64 tx 4 ) exp(72 tx 6 ) . So, we can write exp(8tx 2 ) 2exp(64 tx 4 ) exp(72 tx 6 ) Axt(,) 6 . 1 3exp(8tx 2 ) 3exp(64 tx 4 ) exp(72 tx 6 ) SOLITONS AND NONLINEAR EQUATIONS 103 Let us remark that this solution may be obtained also from (3.5.29). Finally, the solution is given by d 3 4cosh(2x 8tx ) cosh(4 64t ) uxt(,) 2 Axxt (,;) 12 . (3.6.4) dx [3cosh(x 28tx ) 4cosh(3 36t )]2 This solution was derived for t ! 0 , but it makes sense also for t 0 . The solution consists of two waves. The taller wave catches the shorter one, coalesces to form a single wave at t 0 , and then reappears to the right and moves away from the shorter wave as t increasing. The interaction of waves is not linear. The taller wave has moved forward, and the shorter one backward relative to the positions they would have reached if the interaction were linear (Drazin and Johnson). The nonlinear interaction of waves is characterized by the phase shifts. The solitons occur as t orf, and interact in this special way. The asymptotic behavior of the solution for t of, is examined by introducing [ x 16t . The solution is written as 3[[ 4cosh(2 24t ) cosh(4 ) uxt(,) 12 . [3cosh([ 12tt ) cosh(3 [ 12 )]2 Ast of , we obtain the solution 1 uxt( , )| 8sech2 (2 [ ln 3) . 2 This wave is moving with a velocity of 16 units, the amplitude of 8 units, and a ln 3 phase shift of . As t of , we have the wave 2 1 uxt( , )| 8sech2 (2 [ ln 3) , 2 ln 3 having a phase shift of ( ). For K x 4t , the solution becomes 2 34cosh(2)cosh(448)KKt ux(,)t 12 , [3cosh(K 24tt ) cosh(3 K 24 )]2 which yields 1 uxt( , )| 2sech2 ( K ln 3) , t of, 2 and 1 uxt( , )| 2sech2 ( K ln 3) , t of. 2 By approximating we can consider that 11 uxt( , )| 8sech22 (2 [ ln 3) 2sech ( K ln 3), t of, 22 104 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 11 uxt( , )| 8sech22 (2 [ ln 3) 2sech ( K ln 3), t of. 22 The terms exponentially decreasing at t orf were neglected. Note that the solution at f consists of two solitons having the velocities 16 and 14 units, respectively, with the tallest and then fastest soliton moving forward by an ln 3 ln 3 amount M , and the shorter wave moving back by M . The three-soliton 0 2 0 2 solutions are derived from (3.6.1), with N 3 , and uu,0x o , x of, ux( ,0) 12sech2 (x ) . As the previous example, from (3.5.21) we have RkRL(,0) Rk (,0)0 , (1i)(2i)(3i)kkk Tk(,0) , (ikk 1)(i 2)i k 3) and 3 M(x ,i;0) Px12 (tanh ) [5 tanh ( x ) 1]sech(x ) , 3 2 22 M(x , 2i;0) Px3 (tanh ) 15tanh( x )sech (x ) , 33 M(,3i;0)x Px3 (tanh) 15sech()x . The fundamental solutions are 1 fx( ,i;0) [5tanh2 (x ) 1]sech( x ), j 8 11 fx( , 2i;0) tanh(x )sech23 ( x ), fx ( ,3i;0) sech (x ). jj28 According to (3.5.13) we have :R (zt ; ) 12exp(8 t z ) 60exp(64 t 2 z ) 60exp(216t 3 z ) , :L (zt ; ) 12exp( 8 t z ) 60exp( 64t 2 z ) 60exp( 216t 3 z ) , mRL,1(i,0) 12 , mRL,2(2i,0) 60 , mRL,3(3i,0) 60. Further, it is sufficient to study the Marchenko equation (3.5.24) 1 12exp(8txy ) 60exp(64 t 2 x 2 z ) 60exp(216 t 3 x 3 z ) AxytR ( , ; ) f [2exp(8tx '' y ) 12exp(64 t 2 x '' y ) 60exp(216 t 3 x '' 3 zAxxtx )] ( , ''; )dcc 0, ³ R x for x y . Assuming the solutions of the form RR R AR (xyt , , ) L12 ( xt , )exp( y ) L ( xt , )exp( 2 y ) L 3 ( xt , )exp( 3 y ) , SOLITONS AND NONLINEAR EQUATIONS 105 the above equation is reduced to (3.5.27), where ªº1 6exp(8tx 2 ) 20exp(8 tx 3 ) 15exp(8 tx 4 ) Axt( , )«» 4exp(64t 3 x ) 1 15exp(64t 4 x ) 12exp(64t 5 x ) , «» ¬¼«»3exp(216tx 4 ) 15exp(216 tx 5 ) 1 10exp(216tx 6 ) R ªºLxt1 (,) ª 12exp(8tx ) º «» LxtRR(,) Lxt (,) , Bxt(,)« 60exp(64t 2) x » , «»2 « » «»LxtR (,) «60exp(216tx 3 )» ¬¼3 ¬ ¼ detA (xt , ) 2exp(144 tx 6 )[cosh(144 tx 6 ) 6cosh(136 tx 4 ) 15cosh(80tx 2 ) 10cosh(72 t ). On the basis of (3.4.30) we have M (,)xt uxt(,) 12 , Nxt(,) where M (xt , ) 252 12cosh(280t 10 x ) 100cosh(8t 2 x ) 20cosh(224t 8 x ) 160cosh(64tx 4 ) 60cosh(216 tx 6 ) 270cosh(56 tx 2 ) 30cosh(72tx 6 ) 80cosh(208 tx 4 ) 50cosh(152 tx 2 ), Nxt( , ) [cosh(144 t 6 x ) 6cosh(136 t 4 x ) 15cosh(80 t 2 x ) 10cosh(72 t )]2 . Finally, we present the periodic solutions of the KdV equation uuuutxxxx60 , written as a sum of solitons (Whitham 1984). The steadily progressing waves (trains of solitons) are given by ukf 2(2 [) , [ kxZ t , (3.6.5) wheref ()[ satisfies 4 Z f 6(46)6BAf f 3 , A (1 ) , (3.6.6) [[ 6 k 3 with B a constant of integration. For AB 0 , we obtain a single soliton, 2 f s(ech [[0 ),[V0 a constant. A row of these solitons spaced 2 apart f f ([ ) ¦ sech2 ([ 2mV ) , (3.6.7) f leads to an exact solution. Substitution of (3.6.7) into (3.6.6) yield 2 ff f ½24 2 ®¾¦¦sech ([ 2mm V ) sech ( [ 2 V ) B A ¦ sech ( [ 2m V ) . ¯¿f f f 106 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS This identity is true, and we have f 1 1dA (V ) A() 4 , B() . V ¦ 2 V 1 sinh (2jV ) 2dV As referring to the inverse scattering method, we mention that the representation (3.6.7) may be viewed as another instance of the clean interaction of solitons. They are superimposed but keep their identity and do not destroy each other under the nonlinear coupling. Konno and Ito have studied the mechanism of interaction between solitons in 1987, by extending one of independent variables of the equations to complex and by observing how singularities of solutions behave in the complex plane. Following Konno and Ito, we choose t as the complex variable. In the complex t- plane one soliton solution is expressed as sum of poles, which are located equidistantly along the imaginary t- axis. In asymptotic regions two-soliton solution is expressed as a superposition of two separated solitons. Konno and Ito have studied the behavior of singularities during the collision. They introduced an auxiliary function and analyze its zeros which correspond to the poles of the solution at the same places in the complex plane. To illustrate the Konno and Ito method, consider the KdV equation uuuutxx12xx 0 , (3.6.8) which admits the following one soliton solution k 2 1 uxt( , ) sech2 (kxE tG ) , (3.6.9) 42 whereE k 3 . The auxiliary function is w2 uxt(,) log(,)I xt . (3.6.10) wx2 The double poles of the solution correspond with simple zeros of this function. For I(,)xt 0, we have I(,xt* ) 0 , the zeros being symmetrically distributed with respect to the real t- axis. Consider only the upper half plane. For one soliton we have I(,)x tAkxt 1 exp( E ), (3.6.11) whereA expG is a positive constant. We can obtain zeros of I at tknRnIn() t () k i t () k, n 0,rr 1, 2,..., (3.6.12) kx G (2n S 1) t , t . (3.6.13) Rn E Rn E By using the zeros, the soliton solution (3.6.9) is given by k 2 f 1 u . (3.6.14) 2 ¦ 2 En f (()ttkn Therefore, we find that SOLITONS AND NONLINEAR EQUATIONS 107 1. Double poles are equidistantly located in parallel with the imaginary t- axis. 2. Real part tRn ()k depends linearly on x and admits a trajectory of the soliton. 3. Poles located at smaller imaginary parts tIn ()k affect more effectively the soliton amplitude. For two-soliton solution the auxiliary function is I(x ,tAkxtAkxt ) 1 exp( E ) exp( E ) 111222 (3.6.15) EEAkkxt31212exp[( ) ( ) ], 3 with Ej k j ,j 1, 2, and 2 ()kk12 A3 2 AA12. (3.6.16) ()kk12 Asx orf two solitons are separated in each other and locations of the zeros are given by the sum of contributions from two solitons. To increasing x the zeros are classified into two categories: – A zero belonging to one soliton with k1 changes asymptotically into one for another soliton with k2 and vice versa, namely tkIn ()1 | tkIn ()2 . (3.6.17) – The zeros preserve their identity even though they receive effect of nonlinear interaction in such a way as phase shift during interaction. 3.7 Boussinesq, modified KdV and Burgers equations Fermi, Pasta and Ulam studied in 1955 the problem of a dynamical system of n identical particles of unit mass on a fixed end line with forces acting between particles (Dodd et al.). The motion equation of the particle n is QfQQnnnnn 1 fQQ 1 , (3.7.1) whereQn is the displacement of the particle with respect to the equilibrium position, and f(Q) is the interaction force, which may be of the forms f QQQ J D 2 , (3.7.2) or f QQQ J E 3 , (3.7.3) with J a constant, and D , E chosen in such a way that the maximum displacement of Q caused by the nonlinear term is small. Using initial sine-wave data, integrating the equation (3.7.1) with (3.7.2) or (3.7.3) yields to the conclusion that the equipartition of energy criterion failed. The energy is kept in the initial vibration mode and a few nearby modes, without spreading in all the normal modes of vibrations. Using the McLaurin expansion 108 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS ªºw fn()exp()( a «» a fn) , ¬¼wn wheren is now a continuous variable, the equation (3.7.1) becomes (Dodd et al.) ww Qf [(exp( ) 1) Q ] f [(1 exp( ))Q ] , wwnn with Qtn(, ) Qn () t . Expanding the function f as a McLaurin series we obtain ªºªww24QQ11ww QQ2º Qf cc(0) «»«24 ...fc (0) 22 ...» ¬¼¬wwnn12 2! wn wn¼ 2 1 ªº§·wwQQ2 f ccc(0) 3 ... . «»¨¸ 2 3! ¬¼«»©¹wn wn This equation can be written as 244 2 ªºª23wwPl P 1 r wwPP º Pf cc(0) «»« l 24 ...fc (0) 2 l 2...» ¬¼¬wwxx12 2! wx wx¼ 2 1 ªº§·wwPP2 f ccc(0) 3 ... , «»¨¸ 2 3! ¬¼«»©¹wx wx if we note x nl and PQl r , where r is to be determined from the balance of the fourth derivative term with the nonlinear term. For the quadratic nonlinearity given by (3.7.2), it results J f c(0) , 2D f cc(0) and f ccc(0) 0 . Therefore, we choose r 1 , wP and taking ux ,t up to Ol4 , we obtain the Boussinesq equation wx 242 2 ww§·lu 42 2 wu 22¨ Dlu l J u ¸ 2 , (3.7.4) wwx ©¹12 xtw which describes the motion of waves in both directions. In the approximation Ol 2 , (3.7.4) reduces to the linear wave equation. One solution of (3.7.4) is 11 ua 22sech > axtZG@ , (3.7.5) 82D with Z22244 Jal al /12 . (3.7.6) The Boussinesq equation is known either in the form of the system of equations 18 uv , vu uu, txtxxx33x SOLITONS AND NONLINEAR EQUATIONS 109 1 and admits the Hamiltonian v . The system has an infinite number of symmetries and 2 conservation laws, also an infinite number of exact solutions. The Boussinesq equation (3.7.6) can be reduced to a KdV equation if the waves are propagating in only one direction lu33(1)www u(1) u(1) D22lu3(1) J 0 . (3.7.7) 12 w[3 w[ wW Indeed, (3.7.7) is obtained by a scaling transformation on x and t [ Hp x ct, W Hq t , (3.7.8) where H is a small parameter, and by expanding u in powers of H uu H(1) H 2 u (2) ... , (3.7.9) with clp J,1/ 2 and q 3/2. For a cubic nonlinearity given by (3.7.3), it is obtained the motion equation up to the approximation Ol 4 242 2 ww§ lu 43 2 · wu 22¨ Elu l J u ¸ 2 . (3.7.10) wwx ©¹12 xtw This equation may be written under the form of a modified KdV 33(1) (1) (1) luww2 uw u 32lu3(1) 0 , (3.7.11) 3 E J 12 w[ w[ wW for p =1, q =3 and c Jl with wave solutions traveling only in the direction [ 11 ua 22sech > a[ZWG@ , (3.7.12) 6E 2 al33 Z2 . (3.7.13) 24 J With an appropriate changing of variable, the modified KdV equation (3.7.11) may be written under the normalized form 2 uuuuxxx 3 x t 0 . (3.7.14) 2 Miura (1976) has shown that the explicit nonlinear transformation vu i ux , maps solutions of the modified KdV equation into solutions of the KdV equation. f Writing u , the equation (3.7.14) reduces to g 22 g ffxxx f , fgxxx33 f x g xx f xx g x f xxx g g t f gf t 0 , (3.7.15) 110 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS which admits two-soliton solutions ga 211222 expTTTTTT 2 a exp 2 aA exp(2 122 ) 2 aA exp( 12 2 ), fAA 1 exp 2 T12 exp 2 T 2(1 )exp( TT 121 ) exp(2 TT 22 ), (3.7.16) with 3 3 Tii ax at i Gi , Tii ax at i Gi . Whitham obtained in 1984 the periodic solutions for (3.7.14) expressed as sums of solitons. Any steady progressive wave is given by ukf 2()[, [ kx Z t , (3.7.17) wheref ()[ satisfies 1 Z f 2(21)2BAff 3 , A (1) , (3.7.18) [[ 2 k 3 with B a constant of integration. For AB 0 we obtain a single soliton, f s(ech [[0 ). A row of these solitons spaced 2V apart f f ([ ) ¦ sech ( [ 2mV ) , (3.7.19) f leads to exact solution. Substitution of (3.7.19) into (3.7.18) yield 3 ff f ½3 ®¾¦¦sech ([ 2mmBA V ) sech ( [ 2 V ) ¦ sech ( [ 2m V ) . ¯¿f f f f cosh(2jV ) This identity is true, and A 6 , B 0 . Integrating (3.7.18) we ¦ 2 1 sinh (2jV ) obtain a Weierstrass equation with polynomial of four degrees 2 24 f[ CAff (2 1) , (3.7.20) writen as 22222 f[ ()()ffff01 , ff0 ttf . The periodic solutions are expressed as ff 00dn ( f[ ) . (3.7.21) Therefore, (3.7.19) gives the Jacobi elliptic function dn in terms of a sum of sech functions. The modified KdV equation has also an infinite number of conserved 1 quantities. Its Hamiltonian is u2 . 2 Another remarkable equation is the nonlinear diffusion equation uuuutx Gxx , (3.7.22) SOLITONS AND NONLINEAR EQUATIONS 111 known as the Burgers equation. The diffusion coefficient G is a positive and real constant. The Burgers equation (3.7.22) includes nonlinearity and dissipation, leading to a nonlinear version of the heat equation. The simplest solution of (3.7.22) is the Taylor shock wave 1 ua G(1 tanh ( axat G2 )) . (3.7.23) 2 As x of, we have u o 0 , and as x of,uaoG2 . The shock solution is continuous because the term Guxx prevents the tendency of nonlinearity to form discontinuities. For ux(,0) fx ()0 and Go , the solutions of (3.7.22) may be written as a functional ufxut ( ) , (3.7.24) that can be determined by the characteristics method. In 1931 Fay (see Whitham 1984) derived the following solution for Burgers equation ( G 1) f sin nx uxt(,) 2¦ . (3.7.25) 1 sinh nt Another two solutions are derived from the Cole–Hopf transformation 2I u x , (3.7.26) I by taking I to be initially a row of equally spaced functions. We have f I 1 2¦ ( 1)n exp( nt2 )cos nx, 1 11f I ¦exp{ [ (xj ( 1) S ]2 } . (3.7.27) 4St f 4t The last form leads to the saw-tooth profile for small t . Parker found in 1980 (see Whitham 1984) another equivalent form, obtained as a certain superposition of equally spaced shocks of the form (3.7.23) moving with zero speed relative to a basic frame. In the saw-tooth wave the shocks decay, and in order to remain periodic and bounded as x orf, we must add a term proportional to x . The latter adjustment is made from the elementary solution of the Burgers equation, ux / t . Therefore, the Parker exact solution is given by x SSS(2xk) u ¦ tanh , (3.7.28) ttk 2 t N where the summation is interpreted as lim . N of ¦ kN 112 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 3.8 The sine-Gordon and Schrödinger equations We have seen that the sine-Gordon equation is related to the construction of pseudospherical surfaces. In 1967 Lamb obtained the sine-Gordon equation in the analysis of the propagation of ultrashort light pulses (see Coley et al.). He exploited the permutability theorem associated with the Bäcklund transformation to generate an analytic expression for pulse decomposition corresponding to the two-soliton solution. Let us begin with a mechanical pendulum problem, consisted from N identically pendulum of mass M linked by torque springs (Dodd et al.). The total restoring torque exerted by the springs is composed of the torque due to the gravity and the torque due to the torque springs given by *ii Mdgsin M k ( Mi1 MM 2ii1 ) , (3.8.1) whered is the distance of the center of mass from the central axis, g the gravity acceleration constant, Mi is the angle made by the i- th pendulum with the downward vertical, and k is the torque constant. For an array of pendula having the same moment of inertia J , the Newtonian motion equations for the i pendulum is JZi *i , (3.8.2) whereZi is the angular velocity of the i- th pendulum. Inserting (3.8.1) into (3.8.2) we obtain the equation JMdgkM ii sin M ( Mi1 MM 2ii1 ) . (3.8.3) To obtain a continuous model for the system of pendula, a limiting process is considered, by introducing new space and time variable Mdg x Mdg X , Tt , kh J whereh is the distance between pendula. The equation (3.8.3) is reduced to the sine- Gordon equation MMTT XX sin M 0 . (3.8.4) Another form of sine-Gordon equation is the nonlinear version of the known Klein– Gordon linear equation from the field theory, derived by Skyrme in 1958 (Perring and Skyrme) 2 IIxx tt m I, (3.8.5) wherem is a constant, namely 2 IIxx tt m sin I. (3.8.6) For new variables [ mxvt J() G and J2 (1 Q21 ) , the equation (3.8.6) becomes I[[ sin I. (3.8.7) SOLITONS AND NONLINEAR EQUATIONS 113 Multiplying the equation (3.8.7) by I[ and integrating, we have Ic2 Icos C . Assuming the boundary conditions of the form Io0(mod 2 S ) , the integration I constant C is zero. Using cosI 1 2sin2 , it results from ddI I [ 2 [ dI I [ = ln tan , ³ I 4 2sin2 1 2 or I tan exp[. (3.8.8) 4 It is easy to show from I[[ 2sech [ tanh [, sinI 2sech [ tanh [, that the solution of (3.8.8) is given by I 4arctanexp[mxvt J ( ) G ], (3.8.9) with J22 (1 Q ) 1 . (3.8.10) The solution (3.8.9) is named kink because it represents a twist in the variable I(,)x t , which takes the system from one solution I 0 to an adjacent solution with I 2 S. Using the Bäcklund transform we can derive for the equation sine-Gordon multi-soliton solutions. In section 1.7 we have obtained the two-soliton solution I12 (1.7.23) IIPP II tan 12 2 1tan 2 1 , 44PP21 whereI1 and I2 are two solutions for the sine-Gordon equation (3.8.9) I11 4arctanexp[mxvtJ ( ) G1 ], I22 4arctanexp[mxvtJ ( ) G2 ], generated from I given by (3.8.9), by the Bäcklund transform (1.7.23) of parameters P1 and P2 . If I1 is a solution of the type (3.8.9) and I12 , I22 two-soliton solution generated from I13, then a three-soliton solution I is given by II PP I I tan 31 2 1tan 12 22 . 44PP21 Another equation, exhaustively investigated by both physicists and mathematicians, is the nonlinear SchrCdinger equation (NLS) 114 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS wI w2 I i|EII|02 , (3.8.11) wt wx2 whereI is a complex function. Applying to (3.8.11) the Painlevé analysis with associated conditions Io0 as ||x of, we obtain a traveling soliton solution 211 I abxbaaxexp{i [ (22 )]}sech[ ( bt )] , (3.8.12) E 24 wherea andb are arbitrary constants. The propagation of waves in dielectric wires is described by the NLS equation with an additional dissipative term wI1 w2 I i|IIHI|i2 0 , (3.8.13) wt 2 wx2 whereH is a positive constant. When H 0 the solutions have the form I rntexpi( T ) , (3.8.14) with rx() ctand T()x ct real functions, and cn, real constants. Substitution of (3.8.14) into (3.8.9) yields 1 A Tc ()c , SFc2 2()S , (3.8.15) 2 S with 1 Sr 2 , F()SS 32 2( n cSBSC )2 . (3.8.16) 4 In (3.8.16), BC, are arbitrary integration constants. Analyzing the nature of the roots of the cubic equation in the right-hand side of (3.8.16) 2 , we obtain the soliton solution I Asech D expi E , D A(),xct J 1 1 E cx () n22 A t G, Anc22 2( ) ! 0 , (3.8.17) 2 4 andc, J , G constants. If we attach to (3.8.13) an initial condition of the type (3.8.17) for Hz0 , we obtain ddAVddJT 2, HA 0, 0, 2HA2t . (3.8.18) ddttddtt The amplitude of wave is decreasing with time as A At0 exp(H 2 ) , where A0 is the initial amplitude. As the amplitude decreases in time, the width of the wave 1/ A becomes smaller, in contrast with linear waves for which A At0 exp(H ) . SOLITONS AND NONLINEAR EQUATIONS 115 In the case (3.8.13) when the dissipative term iHI has a bigger weight then the nonlinear and dispersive terms, namely for H!!A2 , the waves are damped. In this case the width of the waves remain unchangeable. For another NLS equation wI w2 I i|II|02 , (3.8.19) wt wx2 by repeating the calculations, we obtain the soliton solutions I rntexpi( T ) , rmk22([ ) 2 sech2k [, (3.8.20) ckktanT[ ( ) 2 tanh [, for any c , and 1 1 [ x ct , nm , km (2c21/2 ) , mc! 2 . 2 2 3.9 Tricomi system and the simple pendulum Consider the following differential system of equations (Halanay, Arnold) yfxyyyiic (,12 , ,...)n, in 1,2,..., , where the functions fi , i 1,2,...,n , are continuous and verify the Lipschitz conditions with respect to yii , 1, 2,...,n , |fyyiknikni (12 , ,.., yccc ,., y )d fyy (12 , ,.., y ,., y ) | Ak ( ykccc yk ) , forkn 1,2,..., . By imposing the initial conditions 0 yxii()0 y, the solutions yii , 1,2,..., n , verify the Volterra equations x yx( ) y0 ftyt [, (), yt (),... yt ()]d t, in 1,2,..., , iii³ 12 n x0 and can be recursively calculated by x yxyftytytyt(mm 1)( ) 0 [,() (), ()m (),... ()m ()]dt , m 0,1, 2,... . iii³ 12 n x0 Consider now the Tricomi problem (Tricomi) yyy12c 3 , yy21c y3 , ymy3c 12y , (3.9.1) 116 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS and the initial conditions y1 (0) 0 , y2 (0) 1, y3 (0) 1 , (3.9.2) dy with 01ddm , yc . The recursive relations become dx x x y(1)mmm ( x ) ytyt () () () ()dt , yx(1)mm () 1 ytyt() () ()m ()dt , 123³ 21³ 3 0 0 x yx(mm 1)() 1 kytyt2 () () ()m ()dt , (3.9.3) 31³ 2 0 or (1) (1) (1) yx1 , y2 1, y3 1, x2 yx(2) , y(2) 1 , 1 2 2! x2 ym(2) 1 , 3 2! x35x yx(3) (1 m ) 6 m, 1 3! 5! x24x ym(3) 13 , 2 2! 4! x24x ymm(3) 13 , 3 2! 4! xx35 yx(4) (1 m ) 12 m ... , 1 3! 5! xx24 ym(4) 1 (1 4 ) ... , 2 2! 4! xx24 ymmm(4) 1 (1 ) ... , 3 2! 4! xx35 yx(5) (1 m ) (1 14 mm 2 ) ... , 1 3! 5! SOLITONS AND NONLINEAR EQUATIONS 117 xx24 x6 ymmm(5) 1 (1 4 ) (42 15 ) ... , 2 2! 4! 6! xx24 x6 ymmmmm(5) 1(1)(1542) ... . 3 2! 4! 6! ()mm () () m In this way we can calculate the approximations yy12,, y 3, m 0,1, 2,... . For m of, the solutions of (3.9.1) turn in xx35 yx (1 m ) (1 14 mm 2 ) o ... sn x , 1 3! 5! xx24 x6 ymmm 1 (14) (14416) 2 ...cn o x , (3.9.4) 2 2! 4! 6! xx24 x6 ymmmm 1 (1 ) (16 44 mm 2 ) ... o dn x . 3 2! 4! 6! From (3.9.1), after a little manipulation, we have d 22yycc yy ( y22 y ) 0 , 11 2 2dx 1 2 d 22my ycc y y ( my22 y ) 0 , (3.9.5) 11 3 3 dx 1 3 and by integrating, we obtain 22 22 yy12 const. , my13y const. (3.9.6) On the basis of (3.9.2), relations (3.9.6) become 22 22 yy12 1 , my13y 1, or sn22xx cn 1, mxsn22 dnx 1. (3.9.7) For arbitrary initial conditions 0 0 0 yy11(0) , yy22(0) , y3 (0) y3 , (3.9.8) the solution of (3.9.1) may be written in a general form n yAcn2 ( x ) . (3.9.9) ii ¦ kiDEi k 1 Consider now the motion of the simple pendulum of mass m and the length l (Figure 3.9.1) described by (Teodorescu) g TZ2 sin T 0 , Z2 . (3.9.10) l 118 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS For small T we have sin T#T, and the equation (3.9.10) admits harmonic solutions 2S of period T Z T Atcos( Z M ) . (3.9.11) The constants are determined from the initial conditions T(0) D, T(0) E. (3.9.12) Figure 3.9.1 The simple pendulum. It is preferable to consider two cases: v Case 1. T(0) 0 , T(0) A E , (3.9.13) l with vyA , the speed of mass for l, and Case 2. T(0) D, T(0) 0 , (3.9.14) with vA the speed of a mass in free downfall from the distance a , without initial speed. The energy theorem gives (V!lcovici et al.) 22 vvA 2( gyl ), vl T, or v2 vgya2 2( ), al A , (3.9.15) 2g y t a , yl cos T, al . (3.9.16) When a !l , we have the harmonic oscillations problem. It results vlA 2g. Since lal , we write al cos D , 0 D S, (3.9.17) where the angle D is the amplitude of the motion. The equation (3.9.15) becomes SOLITONS AND NONLINEAR EQUATIONS 119 T 22 2(coscos)Z T D, cosTt cos D, (3.9.18) or DT T 222 4Z (sin sin 2 ) . (3.9.19) 22 The solution of (3.9.19) and (3.9.13) is T 2arcsin[mt sn( Z )] , (3.9.20) with DE m sin2 ( )2 . (3.9.21) 22Z The solution (3.9.20) may be obtained directly from (3.9.10), noting T T sin mt snZ, cos m cnZt , that yields to T 2cnZmt Z, and 2 2 T 4cn Z22m Zt . The period of (3.9.20) is given by 4ZKT , (3.9.22) where K is the complete elliptic integral. For small values of m we have f 135 (2n 1) Tm 2[1(SZ¦ )2 n ]. (3.9.23) n 1 426 2n Another way to derive the solution (3.9.10) and (3.9.14) is to define T 4arctanUt ( ). (3.9.24) 4tanM (1 tan2 M ) T The identity sin 4M , with M , leads to another form of the (1M tan22 ) 4 equation (3.9.10) UUUUU 22222(1 Z U U)0 , which has the solution UA expiZt . (3.9.25) Thus, the solution of (3.9.10) and (3.9.14) is represented by the real part of expression D T 4arctan(A expi Zt ) , A tan . (3.9.26) 4 This solution is known as the breather solution. The importance of this solution may lie in the fact that its rest energy varies from 16, the rest mass of two solitons, down to zero as Z tends to one. Fora l it results in an asymptotical motion. We have vA 2gl , and the equation (3.9.15) may be written under the form 120 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS T T 222 4cosZ . (3.9.27) 2 For (3.9.13) we have T 4arctan(exp Zt ) S. (3.9.28) This solution can be derived from (3.9.10) and (3.9.13) by the Bäcklund transformation. So, consider uv and so that 1 uv 1 uv ()uv Z sin , (uv ) Z sin . (3.9.29) 22t 22t 1 u Forv 0 it results u Zsin , and 22t 1du u u tC ³ 2log|tan | . (3.9.30) 2sin/2Z 0 u 4 The solution (3.9.28) monotonically increases to S as t of , with 4Z E . When al we have a rotational motion, with vA ! 2gl . The equation (3.9.15) becomes T T 22 4(1sin)Om *2, (3.9.31) 2 with gl()1 a ga 2l O2 ()ZZ22, Z2 , 10!m* !. (3.9.32) l 2 22 0 0 l 2 la Solution of (3.9.31) and (3.9.13) is written as 4Z2 T 2arcsin (snOt ) , 2O E, m*2 . (3.9.33) E2 The solution (3.9.33) may be directly obtained from (3.9.10), by defining sin(T / 2) sn Ot , cos(T / 2) cnOt , that yields T 2dnO Ot and T 2cnsn O2*mtt O O . The rotation period is given by S113 Tmm [1 ( )2* ( ) 2*2 ...] . O224 PART 2 APPLICATIONS TO MECHANICS Chapter 4 STATICS AND DYNAMICS OF THE THIN ELASTIC ROD 4.1 Scope of the chapter The theory of the thin elastic rod occupies an important position in the history of vibration theory. Wallis (1616–1703) and Joseph Sauveur (1653–1716) have observed that a stretched string can vibrate in parts with certain nodes at which no motion takes place, whereas a strong motion takes place at intermediate points, called ‘loops’. The dynamical explanation of this vibration was provided by Daniel Bernoulli (1700–1782) in 1755. He stated the famous superposition principle of the coexistence in the vibrating string of a multitude of small oscillations at the same time. The elastic line in which the resistance of a bent rod is assumed to arise from the extension and contraction of its longitudinal filaments was investigated by James Bernoulli (1667–1748). Daniel Bernoulli suggested to Leonhard Euler (1707–1783) that the differential equation of the elastica could be found by making the integral of the square of the curvature taken along the rod a minimum. Euler obtained on this suggestion the differential equation of the curve. J. L. Lagrange (1736–1813) and Lord Rayleigh (1842–1919) anticipated in their works the fundamental ideas of the modern topological and perturbation method foundered later by Poincaré and Lyapunov. We must mention that the problem of the bending and twisting of thin rods was solved in an elegant analytical fashion by Love in 1926. He made a classification of the form of the rod, which is inflexional elastica and non-inflexional elastica. Since the thin elastic rod is an example of a solitonic medium, it is interesting to see that the inflectional elastica can be reobtained by using the soliton theory. Due to the large displacements in the fundamental equations, even if the strains are small, the nonlinear effects yield to the soliton solutions. The study of solitons in the thin elastic rod is an exciting branch in mechanics. Besides the strings and rods there are many physical problems that can be treated as 122 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS long thin elastic rods. In biology there are many 1D media such as DNA, RNA and D - helix of protein. In this chapter the basic laws of equilibrium and motion for a thin elastic rod are studied and solved, following the Tsuru and Munteanu works. The localized solutions are expressed by elliptic and hyperbolic functions. The solutions expressed by the hyperbolic functions are solitons. The rod deviates from a plane and has a 3-dimensional structure, changing its form as the torsion angle increases. This chapter is referred to the publications of Love (1926), Tsuru (1986, 1987), Munteanu and Donescu (2002), Antman (1974) and Antman and Liu (1979). 4.2 Fundamental equations Let us consider a thin elastic, homogeneous and isotropic rod of length l, straight and having a circular cross section of radius a l in its natural state. External forces and couples fix the ends of the bar. We suppose the rod deforms in space by bending 3 and torsion. The rod occupies at time t 0 the region :9 R . After motion takes place at time t , the rod occupies the region :()t . We know the motion of the rod between t 0 and t t1 if and only if we know the mapping St(0, ), t [0, t1 ] , (4.2.1) which takes a material point in :0 at t = 0 to a spatial position in :()t at t = t1 . The mapping (4.2.1) is single valued and possess continuous partial derivatives with respect to their arguments. The position of a material point in :0 may be denoted by a rectangular fixed coordinate system X { (,,)XYZ and the spatial position of the same point in :()t , by the moving coordinate system x { (,xyz ,). Following the current terminology, we shall call X the material or Lagrange coordinates and x the spatial or Euler coordinates. The origin of these coordinate systems is lying on the central axis of the rod. The motion of the rod carries various material points through various spatial positions. This is expressed by (Truesdell and Toupin, ùoós and Teodosiu, ùoós) xfXti i (,),1,2,3 . (4.2.2) We take s to be the coordinate along the central line of the natural state. The orthonormal basis of the Lagrange coordinate system is denoted by (,eee123 , ) , and the orthonormal basis of the Euler coordinate system by (,ddd123 , ) . The basis ^`dkk ,1,2, 3 is related to ^`ekk ,1,2, 3 by the Euler angles T,\ andM . These angles determine the orientation of the Euler axes relative to the Lagrange axes (Tsuru) de11 ( sin \ sin M cos \ cos M cos T ) \M\MTTM(cos sin sin cos cos )ee23 sin cos , STATICS AND DYNAMICS OF THE THIN ELASTIC ROD 123 de ( sin \ cos M cos \ sin M cos T ) 21 (4.2.3) \M\MTTM(cos cos sin sin cos )ee23 sin sin , dee31 sinT\T\T cos sin sin 2 cose3 . The Z-axis coincides with the central axis. The plane ()xy intersects the plane (XY) in the nodal line ON (Figure 4.2.1). The motion of the rod is described by three vector functions 3 Ru R(,) sto rst (,), d12 (,), st d (,) st E . (4.2.4) The material sections of the rod are identified by the coordinate s. The position vector rst(,)can be interpreted as the image of the central axis in the Euler configuration. The functions ds12(,),tdst (,)can be interpreted as defining the orientation of the material section s in the Euler configuration. The function dst312(,) dstdst (,)u (,), (4.2.5) represents the unit tangential vector along the rod and can be expressed as d3 (sinT\ cos ,sinT\ sin ,cos T ) . We introduce the strains yyy12,,3 by rydc kk, (4.2.6) where ' means the partial differentiation with respect to s. Since ^`dkk ,1,2, 3 , is orthonormal, there is a vector u such as dudkc uk . (4.2.7) Figure 4.2.1 The Euler angles T\, and M . The components of u with respect to the basis ^`dk are 1 ue ddc, (4.2.8) kklml2 m where eklm the components of the alternating tensor. The relations (4.2.8) become 124 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS udddddd131213222332 ccc3 , udddddd211311232133 ccc3 , udddddd32111221223 ccc13 , where dijij ,1, 2,3 , are the components of vectors di , i =1,2,3, given by (4.2.3). Substitution of (4.2.3) into the above relations gives u1 TccsinM\ sin T cos M, u2 TcccosM\ sin T sin M , (4.2.9) u3 Mcc\cos T . These functions measure the bending and torsion of the bar. The functions ukk ,1,2, 3 , can be interpreted as the components of the angular velocity vector (the variation with respect to s) of the rotational motion of the moving system of coordinates relative to the fixed system of coordinates. If we substitute the differentiation with respect to s with the differentiation with respect to time we will obtain the components of the angular velocity vector defined by (4.2.9). The functions u1 and u2 represent the components of the curvature of the central line denoted by N corresponding to the planes()yz and ()xz 222 2 22 N uu 12 T\ccsin T, (4.2.10) andu3 is the torsion of the bar denoted by W u3 W Mcc \cos T . (4.2.11) In this way we consider the rod is rigid along the tangential direction and the total length of the rod l is invariant, the ends being fixed by external forces. The full set of strains of the rod is ^`ykk,u . In the natural state d3 coincides with rc , and dk are constant functions of s. The values of the strains in the natural state are yy12 0, y 3 1, uk 0 . (4.2.12) In the following we assume that extensional and compression strains have the values yy12 0, y 3 1 and focus only on the bending and torsion of the rod. The link between the position vector rx (,yz ,) and unit tangential vector d3 is obtained from the first two relations of (4.2.12) and (4.2.6) rc d3 . (4.2.13) From (4.2.13) we obtain s rd ds , (4.2.14) ³ 3 0 STATICS AND DYNAMICS OF THE THIN ELASTIC ROD 125 or s s x()ss ³ cossind\T, ys() ³ sinsind\Ts, 0 0 s zs() ³ cosdT s. (4.2.15) 0 To characterize the position of ends of the rod we introduce the vector D of the components x(),(),()LyLzL l Dd ds . (4.2.16) ³ 3 0 The elastic energy of the deformed rod U is composed of the bending energy and the torsional energy (Landau and Lifshitz, Solomon) ACll U ³³NW2ds 2ds , (4.2.17) 2200 whereN and W are given by (4.2.10) and (4.2.11). The quantities A and C are the bending stiffness and respectively the torsional stiffness related to the Lam- constants O , P by 11 AaECa S4 , SP4 , (4.2.18) 42 POP(3 2 ) where E is the Young's elastic modulus, and a is the radius of the cross OP section of the rod. The elastic energy can be written in the form by using (4.2.10) and (4.2.11) ACll U ³³(sin)d(cos)T\TM\Tcc222s cc 2ds . (4.2.19) 2200 To write the equilibrium equations, the variation of the elastic energy U with respect to TM, and \ is considered. THEOREM 4.2.1 The exact static equilibrium equations of the thin elastic rod with the ends fixed by the external force F O with O (,OO123 , O ) are given by AC(\TTTM\T\Tcccccc2 sin cos ) ( cos ) sin (4.2.20) O123cos T cos \O cos T sin \O sin T 0, w [sin(AC\TM\TTccc2 cos)cos] ws (4.2.21) O12sin T sin \ O sin T cos \ 0, 126 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS w [(M\TC cc cos)]0 . (4.2.22) ws The end couples at s 0 and s 1 are Mii (0), 1,2,3 , and respectively Mli (), i 1,2,3, where M1 ()sA Tc , the couples with respect to the nodal line ON, (4.2.23) 2 Ms2 ( ) A\ccc sin T\ C ( cos TM )cos T, the couple with respect to Z-axis, (4.2.24) Ms3 () C (Mc cos T\c ), the couple with respect to z-axis. (4.2.25) Proof The unknowns of the problem are Euler angles. The vector is assumed to be known. The position vector rxyz(, ,) is related to d3 by (4.2.13). To take this constraint into account we introduce the functional U OD , (4.2.26) where O (, O123 O , O ) is the Lagrange multiplicator, and D is defined by (4.2.16). Applying the variational Hamilton principle, the variation of is given by G G U OGD 0 , (4.2.27) whereGD is the variation of the end positions. We write (4.2.27) under the form l G G³ Ls(, T Tccc ,, \ \ , M , M )d 0, 0 where the Lagrange function L is AC L (sin)(cos Tcc222 \ T \ c TM c)2 (4.2.28) 22 O12sin T cos \O sin T sin \O3 cos T. The variation of the functional with respect to T, M and \ is l wwLL w L w L w L w L G ³ () GT GTcc G\ G\ G\ GMc d0s . 0 wT wTcc w\ w\ wM wM www Observing that GTccc GT,, G\ G\ GM GM , and integrating by parts the wws ssw wwwLLL terms GTcc,, G\ GMc , we obtain wTcc w\ wM STATICS AND DYNAMICS OF THE THIN ELASTIC ROD 127 l ll wwLL w L G GT G\ GM wTccc0 w\00 wM ll §wLL w§· w ·§· w L w§· w L ³³¨¸¨¸ GTdds ¨¸¨¸ G\s (4.2.29) 00©¹wT wss©¹ wTcc©¹w\ w©¹ w\ l §·wwwLL§· ³ ¨¸¨¸ GMd0.s 0 ©¹wM ws ©¹ wMc Taking into account that GT(0) GT (l ) 0 , G\(0) G\ (l ) 0 , GM(0) GM (l ) 0 , (4.2.30) the first three terms in (4.2.29) are vanishing and we obtain the Lagrange equations wwwLL§· wwwLL§· wwwLL§· ¨¸ 0, ¨¸ 0, ¨¸ 0. (4.2.31) wT ws ©¹ wTc w\ ws ©¹ w\c wM ws ©¹ wMc The conditions (4.2.30) express that the functions T GT,, \ G\ M GM have the same values at s = 0 and s = l. The nonvanishing integrals must be zero for any variations GT,, G\ GM and this is possible only if the integrands are identically zero. Substituting (4.2.28) into (4.2.31) we obtain the differential equations (4.2.20)– (4.2.22). If F (,FFF123 , ) is the force applied to the rod's ends, with Fii ,1,2, 3 , the components of the force with respect to the fixed coordinate system (,,)X YZ , then this force is related to O by wU F O. (4.2.32) wD Therefore,Ȝ represents the external force that fixes the ends of the rod. This force is supposed to be known. The couples M (,MMM123 )can be determined from l G ³{(TS (4.2.20))GT (TS (4.2.21))G\ (TS (4.2.22))GM }ds 0 2 l TGT\TM\TTG\M\TGM[AAc [ c sin C ( cc cos )cos ]C (cc cos ) ]0 , (4.2.33) where TS are left-side parts of equations (4.2.20)–(4.2.22). Into equilibrium the integrand in (4.2.33) is zero G [AA Tcc GT [ \ sin2 T C ( M cc \ cos T )cos T ] G\ (4.2.34) l M\TGMC(cos)].cc 0 From here we derive the couples at the ends of the rod with respect to ON and Z and z axes w wU MA |||| Tc , 10wTs or l wT s 0 or l s 0 or l 128 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS w wU MA | | |\T\TMTccc sin2 C ( cos )cos | , 20w\s or l w\ s 0 or l s 0 or l w wU MC |||(cos)| MT\cc. 30w\s or l w\ s 0 or l s 0 or l The equilibrium equations (4.2.20)–(4.2.22) are coupled nonlinear ordinary differential equations in unknown Euler angles. Next, we see that the equation (4.2.22) can be solved M\cc()ss ()cos() T s D, with D an integration constant. From the torsion definition (4.2.11) we conclude that D W. So, the above relation becomes M\cc()ss ()cos() T s W. (4.2.35) With (4.2.35), equations (4.2.20) and (4.2.21) can be written as AC(sincos)\cc2 T TTcc W\ sincoscos TO T \ 1 (4.2.36) O23cos T sin \O sin T 0, AC(\T\TTWTO\O\cc sin 2 c c cos )c 12 sin cos 0 . (4.2.37) We introduce 22 O1 O O12 O , \1 \Sarctan , (4.2.38) O2 and write O\O\cos sin O\ sin , 12 1 (4.2.39) O\O\12sin cos O\ cos1 . Adding (4.2.36) multiplied by 2 Tc and (4.2.37) multiplied by (2 \1c sinT ) we obtain 22 2 2 2 AA\11c(sinT ) c (\ cc ) sinT A ( T cc ) O 3 (cos T ) c O2 sin \11 (sin TO )cc 2 sin T (sin \ ) 0. Dividing by 1/2 we have A [(Tcc222\ sinTO ) sin T sin\ O cos T ]c 0 . (4.2.40) 2 113 Integrating (4.2.40) with respect to s we find the bending energy density of the thin elastic rod A N2 Osin T sin\ O cos TC , (4.2.41) 2 13 0 STATICS AND DYNAMICS OF THE THIN ELASTIC ROD 129 with C0 an integration constant. In the case O (0,0,O3 ) the equilibrium equations (4.2.35)–(4.2.37) become Mcc()ss \ ()cos()T s W, (4.2.42) 2 AC(\cc sinTTTW coscc )\ sinTO3 sin T 0 , (4.2.43) AC(sin2\ccT \ cTTWT c cos)c 0. (4.2.44) To write the motion equations, let us introduce the inertia of the rod characterized by the functions Rs o(UUU00 As )( ),( 0 I1 )( s ),(02 Is ( )f (0, ) , (4.2.45) where ( U 0A0) is the natural mass density per unit length, A0 the area of the cross section, ( U0 I1) the principal mass moment of inertia around the axis which is perpendicular to the central axis and ( U0 I2 ) the principal mass moment of inertia around the central axis. We suppose to have Sa44Sa U AakIU S 2U ,, U U kI U U , (4.2.46) 00 0 1 1042 0 2 20 0 where U0 is the mass density per unit volume, and I12, I geometrical moments of inertia around the axis, which is perpendicular to the central axis and respectively around the central axis. The kinetic energy K of the rod is a sum between the energy of the translational motion K1 , the energy of the rotational motion of the tangential vector K2 and the energy of the rotational motion around the central axis K3 (Tsuru) K KKK123, (4.2.47) with U l K rs2d , (4.2.48) 1 ³ 2 0 k l k l K 1 ds 2d 1 (::22)ds , (4.2.49) 2 ³ 3 ³ 12 2 0 2 0 k l K 2 :2ds , (4.2.50) 3 ³ 3 2 0 where the dot represents the differentiation with respect to time, and ::(,123 : , : )is the vector of angular velocity of rotation :1 \ sinT cosM T sinM , :2 \ sinT sinM T cosM , (4.2.51) :1 \cos TM . 130 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS These relations are analogous to (4.2.9). Using (4.2.7) we get for K2 and K3 k l K 1 (sin\TT 22 2 )ds , (4.2.52) 2 ³ 2 0 k l K 2 (cos\TM )d2 s . (4.2.53) 3 ³ 2 0 We prove now the following theorem: THEOREM 4.2.2 The exact set of motion equations of the thin elastic rod with the ends fixed by the force F =-O with O (, O123 O , O )are Ur Oc 0 , (4.2.54) 2 kk12(\TTTM\T\T sin cos ) ( cos ) sin AC(\cc2 sinTTT coscccc ) (M \ cosT )\ sin T (4.2.55) O123cos T cos \ O cos T sin \ O sin T 0, w \TM\TT{sin(kk2 cos)cos} wt 12 w \{sin(ACccc22 TM\TT cos)cos} (4.2.56) ws O12sin T sin \O sin T cos \ 0, ww M\TM\TkC(cos)( cc cos) 0 . (4.2.57) 2 wwts Proof The unknowns are the force vector Ȝ and Euler angles. The position vector is related to Euler angles through the constraint (4.2.13). The kinetic energy K is given by LL L U kk2 K rs+d2dd(cos1 s2 \TM)d2 s. (4.2.58) ³³3 ³ 22002 0 We introduce the Lagrangian LKU , (4.2.59) and the action T I ³ dtL . (4.2.60) 0 To take account of the constraint (4.2.13) we introduce a new functional Tl IILI dd( tsr-dc )O. (4.2.61) c ³³ 3 00 STATICS AND DYNAMICS OF THE THIN ELASTIC ROD 131 We write this functional under the form Tl U kk AC Itsr+d++ dd[2212 (cos)\TM222NW +rd()c O] , c3³³ 3 0 0 22 2 2 2 or Tl I dd(t sLr,r,r, cc , , ), c ³³ 00 whereȢ is the vector of Euler angles Ȣ(,MT\ , ) and L the Lagrangian U 22kk12 2A 222 Lr+d+ 3 (cos\T +M ) (Tcc\ sin)T 22 2 2 (4.2.62) C M\T(cos)().cc2 +r c O d 2 3 The variation with respect to r and Ȣ gives Tl wwLL w LL w w L w L GItsrr dd( G Gccc G r G G G )0. c ³³ 00 wwrrcc w r www wwww We see that Grrrrcc G,,, G GG GG G , and by integrating wwws tswt wwLL through parts the terms GGrrc, etc., we obtain wwrrc TT llTTll wwwwLLLL GIrc GGGG r wrrcc00 w w w 000000 Tl §wLL w§· w w §· w L · G³³ddts¨¸¨¸ ¨¸r 00 ©¹wwwrsr©¹c ww tr ©¹ (4.2.63) Tl §·wwwwwLL§· §· L G³³ddts¨¸¨¸ ¨¸ 0. 00 ©¹wwwst©¹ c ww ©¹ According to GrrT(0,0) G (0, ) 0 , Grl(,0) G rlT (, ) 0, GȢȢ(0,0) G (0,T ) 0 , GȢȢ(,0)llT G (, ) 0, (4.2.64) the first four terms in (4.2.63) are vanishing. The condition for integrands to be identically zero lead to the Lagrange equations wwwwwLL§· §· L wwwww LL§· §· L ¨¸ ¨¸ 0, ¨¸ ¨¸ 0. (4.2.65) wwwrsr©¹cc ww tr ©¹ wwwww s©¹ t ©¹ 132 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS By substituting (4.2.62) into (4.2.65) we obtain the motion equations (4.2.54)– (4.2.57). Differentiating both sides of (4.2.54) with respect to s, and using (4.2.13) the equation (4.2.54) becomes Ud3 Occ 0 . (4.2.66) The motion equations (4.2.54)–(4.2.57) are coupled nonlinear partial differential equations in unknown Euler angles and the vector function O which characterizes the external force applied to the ends of the rod to maintain it fixed. We have to supplement the motion equations with initial conditions 2 O(,0)ssvds O03 () UO (,0)(0,0,3 ), T(,0)s T0 ()s , \(,0)s \0 ()s , (4.2.67) M(,0)s M0 ()s . 4.3 The equivalence theorem This section focuses on the relation between the equilibrium equations and the motion equations of the thin elastic rod (Tsuru). The purpose is to determine the conditions when the motion equations can be reduced to the equilibrium equations. In connection to this, the following theorem holds: THEOREM 4.3.1 GivenOT\,, and M as functions only of the variable [ s vt and suppose that O ]d3 O(0,0,3 ) , (4.3.1) 2 with ] Uv and d3 (sinT\T\T cos ,sin sin ,cos ) . In these conditions the motion equations (4.2.54)–(4.2.57) are equivalent to the equilibrium equations (4.2.20)– (4.2.22) for 22 A okv12 A,, C o[o kv C s. (4.3.2) Proof We note by prime the differentiation with respect to the new variable [ s vt . The system of equations (4.2.54)–(4.2.57) become Uvr2 cc O c , (4.3.3) 22 2 kv12(\TTTM\T\Tcc sin coscccc )kv ( cos ) sin \AC(sincos)(cc2 T TTM\T\Tcccc cos)sin (4.3.4) O123cos T cos \ O cos T sin \ O sin T 0, STATICS AND DYNAMICS OF THE THIN ELASTIC ROD 133 w \TM\T{sin(cos)cosvkccc2 vk T w[ 12 22 \ACcccsin T ( M\ cos T )cos TO }1 sin T sin \ (4.3.5) O2 sin T cos \ 0, w M\TM\Tkvv[cos(cos)]cc C cc 0 . (4.3.6) 2 w[ Rearranging the terms in (4.3.3)–(4.3.6) we have Uvr2 c O cˆ , (4.3.7) 22 ()(sincos)Akv\TTT1 ccc 2 ()(cos)sinCkv2 M\T\Tcc c (4.3.8) O123cos T cos \O cos T sin \O sin T 0, 2 ()(sin2cosAkv\T\TT1 cc c c ) 2 ()(cos)CkvTM\T2 cc c (4.3.9) O12sin \ O cos \ 0, M\cc()ss ()cos() T s W, (4.3.10) with cˆ an integration constant. Taking into account (4.2.13) the equation (4.3.8) becomes 2 Uvd3 O cˆ , or 222 Ȝ (Uvcvcv sin T cos \ˆˆ12 , U sin T sin \ , U cos Tcˆ3 ) . (4.3.11) From (4.3.1) we have Ȝ (] sinT\ cos ,] sinT\ sin ,] cosTO3 ) . (4.3.12) It can be seen that equations (4.3.12) and (4.3.13) are equivalent if 2 ] Uv , cˆ (0,0,O3 ) . (4.3.13) With these conditions, the motion equations (4.2.54)–(4.2.57) become ()(sincos)Akv\TTT22ccc 1 (4.3.14) 2 ()(cos)sinsinCkv23 M\T\TOTcc c 0, ()(sin2cosAkv\T\TT2 cc c c ) 1 (4.3.15) 2 ()(cos)0Ckv 2 TM\cc c T , M\cc()ss ()cos() T s W. (4.3.16) 134 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS Hence, the equations (4.2.54)–(4.2.57) coincide with equilibrium equations (4.2.35)–(4.2.37). We note that the quantity O3 is known. The constant v, which defines the variable [ can be determined from (4.2.67) 1 . 4.4 Exact solutions of the equilibrium equations Consider the case when the force O (0,0, O3 ) with O12 O 0 is parallel to the Z- axis of the Lagrangian system of equations. Let us consider the equilibrium equations (4.2.42)–(4.2.44) and introduce O12 O 0 2 AC(\cc sin T cos TTcc ) W\ sin TO3 sin T 0 , (4.4.1) AC(sin2\T\TTWTcc c c cos)c 0, (4.4.2) M\cc()ss ()cos() T s W. (4.4.3) Multiplying both sides of (4.4.2) with sin T we get AAsin22T\cc (sin T ) c \ c C W (cos T ) c 0 , or AC(sin2 T\cc ) W (cos T ) c 0 , integrating we have ACsin2 T\c W cos T E 0 , (4.4.4) with E an integration constant. In the virtue of (4.4.4) we obtain CWTEcos \c . (4.4.5) Asin2 T Substituting (4.4.5) into (4.4.1) and multiplying the resulting equation with 2Tc we obtain by integration (CCWTE cos )2 WTEcos ACTc2 2 W cos T O2 cos TJ 0 , AAsin22TTsin 3 with J an integration constant. The above equation can be written as C 22WTEcos 2 2 ATc2 O2cos TJ 0. (4.4.6) Asin2 T 3 Substituting u cosT into (4.4.6) we obtain the differential equation 22222 2 1WEuCu½ uuc ®¾2 OJ2 3 , (4.4.7) AAu¯¿(1 ) or STATICS AND DYNAMICS OF THE THIN ELASTIC ROD 135 1 ufuc2 (), (4.4.8) 2 11C 22WE12 fu() [ O u32 ( J )u O u ( J )]. A 3322A A We have obtained a Weierstrass equation (1.4.14) with a polynomial of third order. The torsion W and the integration constants J and E are determined from the boundary conditions \(0) \ (l ) \0 , T(0) T (l ) T0 , W(0) W (l ) W0 . (4.4.9) The qualitative nature of the solutions of (4.4.8) for arbitrary values of the constants can be studied by elementary analysis. We write the equation (4.4.8) in the form 1 uaubuaucfuc232 {(), (4.4.10) 2 with O 11C 22WE2 ab 3 z0, ( J ), c ( J ) . (4.4.11) A 22AA AA We are looking for general bounded waves of permanent form. We have uc2 t 0 . So, u varies monotonically until uc vanishes. The graph of f for different values of the constants a, b and c has six possible forms (Figure 4.4.1). Since ufc2 ()u t 0, solutions will occur only in the intervals shaded in the figure. If u1 is a simple zero for f, from Taylor's formula we have 2 fu() fu (111 ) ( u u ) fc ( u ) O {( u u1 )}, or, because fu()1 0 2 2 ufuuuOuucc 2()(11 ) {( 1 )}, foruuo 1 . (4.4.12) Therefore, 1 us() us () ( s s )() ucc s ( s s )2 uc () s O {( s s )3}, (4.4.13) 111112 1 2 2 where us()1 u1 . From ufc 2()u we have usc ()11 2(())0 fus , and usc()1 0 . Differentiating both sides of ufc2 2()u , it results uscc() fus c (()) and uscc()1 fu c ()1 . From (4.4.13) we get 1 uu fussc()( )23 Oss {( )}, for s o s . (4.4.14) 1112 1 1 136 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS It follows that u has a local minimum or maximum u1 at s1 , as fuc()1 is positive or negative, respectively. Similarly, if u1 is a double zero of f we have 1 fu() fu ()( u u ) fcc () u ( u u )23 fc () u O {( u u )}, 111112 1 or 223 ufccc ()(uuu11 ) Ouu {( 1 )}, foruuo 1 . (4.4.15) The quantity uc is real only if fucc()1 ! 0 (Figure 4.4.1b) and in this case we have ufccc ()(uuu1 1 ). (4.4.16) Relation (4.4.16) yields uc fucc()1 , uu 1 and integrating with respect to s we have ln(uu11 ) r fuscc ( ) ln C , where uu1 C exp(r fuscc (1 ) . (4.4.17) Figure 4.4.1 Schematical graphs of the function f(u). b Thus,uuo as s ofB . There is only one possibility for a triple zero u 1 1 3a (Figure 4.4.1f). The exact solution is b 2 us() 2 , (4.4.18) 3(a s s0 ) STATICS AND DYNAMICS OF THE THIN ELASTIC ROD 137 where s0 is an arbitrary constant. This solution is unbounded at s s0 . Consider now the cases a, d, e, and the right-hand part of the case c shown in Figure 4.4.1. If uc !!!of0 , then f 0 for all s 0 and u , as s of. If uc(0) 0 , then u will decrease until it reaches u u11. In this case u is a simple zero and so uc changes sign and once again u of as s of. So, for these four cases we have unbounded solutions. Let us see the case from Figure 4.4.1b. The solution u has a simple zero at u3 and a double zero at u1 . The solution has a minimum at u u3 and attains uu 1 as s orf. Here we have a solitary wave solution with a amplitude uu31 0 (Figure 4.4.2a) 22 uu 323( u u )sin I u 323 ( u u )(1 cn I) 22 uuu323()(1cn)()sn I uuu323 I ||a uuu()cn(()(),) 2 uussm . 223 2 13 3 Finally we consider the cases in Figure 4.4.1c. The solution u has simple zeros, that is a local maximum at u23 and a local minimum at u . Thus uc changes sign at these points and since the behavior near them is algebraic, consecutive points u u2 , u u3 , will be a finite distance apart. The solution will oscillate between u2 and u3 with a finite period (Figure 4.4.2b). The period can be determined from uc 2()f u as uu22ddu u 22 . (4.4.19) ³³uc 2()fu uu33 We have us22()dduus us22 () c s2 u 2 d u ds . ³³³³uucc 2()fu us33() us33 () s3 u 3 From the above formula we have implicitly the solution u du ss 3 , (4.4.20) ³ 2()fu u3 r where us()3 u3 . The sign + corresponds to the case uc ! 0 and the sign – to the case uc 0. Solutions (4.4.20) are known as the cnoidal solutions because they are described by the cosine and sine Jacobean elliptic functions (Freeman). These solutions were found by Korteweg & de Vries in 1895. Finally, the single bounded solutions of the equation (4.4.8) are the cnoidal solutions (the solitary solution is a particular case of the cnoidal solution). We can write the solution (4.4.20) as (Abramowitz and Stegun) 138 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS u du ss 3 , (4.4.21) ³ 2au ( u )( u u )( u u ) u3 r123 with uuu32 1 three distinct and real roots of the equation f(u) =0. Figure 4.4.2 a) Soliton solution, b) Cnoidal solution. The integral (4.4.21) may be reduced to an elliptic integral of the first kind ||a vuus ()(s ) , (4.4.22) 2 13 3 uu with m 23,0 ddm 1 , the modulus of the elliptic functions. The solution is uu13 22 uu 323()sin()(1cn) u u I u 323 u u I 2 2 uuu323()(1cn)()sn I uuu323 I (4.4.23) ||a uuu()cn(()(),) 2 uussm . 223 2 13 3 We have I dT v . (4.4.24) ³ 2 0 1sinTm The period of cos I is 2S , so the period of the function cn is S /2 dT 44 K(m). (4.4.25) ³ 2 0 1sinTm In fact K(1) f , then the period of the function sech v is infinite. Parameters u2 andu3 determine the amplitude of the cnoidal functions. The parameter s3 determines the phase. The period of the solution u is STATICS AND DYNAMICS OF THE THIN ELASTIC ROD 139 2 2()Km . (4.4.26) uu13 The case of three distinct and real roots. We will prove the following theorem in the case when the equation fu() 0 has three distinct and real roots. THEOREM 4.4.1 Given uu32 u1 the distinct and real roots of the cubic equation fu() 0, the equilibrium equations (4.4.1)–(4.4.3) have a unique solution for the Euler angles ||O uu ()cn(()(),) u u2 3 u u ssm 223 2A 13 3 (4.4.27) 2 uuu223( )cn [ wssm ( 3 ), ], uu2 3 ||O3 with m and wu ()13u , uu13 2A 1 EC W uu \ {[(),, 3ws s23 m] 4Aw22 11uu3 3 3 (4.4.28) EC W uu23 3[(ws s3 ), , m ]}, 11uu33 W()CA 1 EWC uu M s {[(),,3ws s23 m] Au4Aw22 113 u 3 3 (4.4.29) EC W uu23 3((ws s3 ), , m )}, 11uu33 where 3(,,x zm )is the normal elliptic integral of the third kind x dy 3(,,xzm ) . (4.4.30) ³ 2 0 1sn(,) zym Proof The solution of (4.4.7) is given by (4.4.23) that implies (4.4.27). In the virtue of (4.4.3) we have CuWE \c . (4.4.31) A(1 u2 ) Decomposing in simple fractions, we rewrite (4.4.31) as ECC W E W \c . (4.4.32) 2(1)A uAu 2(1) From (4.4.27) we have 140 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 2 11uuuuwss 323 ()sn[(), 3m] , 2 11uuuuwss 323()sn[(), 3m] , (4.4.33) or 1 u uu23 2 1sn[() ws s3 ,] m , 11uu33 1 u uu23 2 1sn[() ws s3 ,] m. (4.4.34) 11uu33 We introduce (4.4.34) into (4.4.32) and integrate with respect to s. The solution (4.4.28) is straightly obtained. Substituting (4.4.31) into (4.4.3) we get W()CAu2 E uAW Mcc W\cos T . (4.4.35) Au(1 2 ) Decomposing in simple fractions, we have for (4.4.35) W()CA EW C E CW Mc . (4.4.36) A 2(1)Au 2(1) Au Similarly, the solution (4.4.29) is obtained. We now prove another theorem. THEOREM 4.4.2 In conditions given by the theorem 4.4.1, the components of the position vector are xR sin(\' ) R sin ), (4.4.37) yR cos(\' ) R cos), (4.4.38) ()uu zu s13 E[am(wss ( )), m ] , (4.4.39) 1 w 3 with 1 22 2 RA 2|OWJE3 | uCA, (4.4.40) O3 §·A ' arctan¨ 2fu ( ) ¸ , (4.4.41) ©¹EuC W E ) s 3CwsCm(,,), (4.4.42) 2A 01 where f(u) is given by (4.4.8) and C01 , C are constants, and STATICS AND DYNAMICS OF THE THIN ELASTIC ROD 141 I v Eamv( (),)(,) m Evm ³³ 1TT m sind22 dn(,)dym y, (4.4.43) 00 is the elliptic integral of the first kind. Proof The components x andy of the position vector are obtained from (4.2.15)1,2. Multiplying (4.4.1) by sin \ and (4.4.2) by cosT cos \ and subtracting we have d [TAAcc sin \\ sin T cos T cos \W C sin T cos \ ] ds (4.4.44) O33sin T sin \ O d32 , whered32 is the y component of the unit tangential vector. Multiplying (4.4.1) by cos \ and (4.4.2) by cosT\ sin and summing we have d [AAT\\TT\WT\cc cos sin cos sin C sin sin ] ds (4.4.45) O33sin T cos \ O d31 , whered31 is the x component of the unit tangential vector. Integrating with respect to s (4.4.44) and (4.4.45) we have s 1 x dsd T [ Acc cos \\T A sin cos T\WT\ sinC sin sin ] x, (4.4.46) ³ 31 0 0 O3 s 1 yds d [T\\TT\WT\ Acc sin A sin cos cosC sin cos ] y, (4.4.47) ³ 32 0 0 O3 with x00, y integration constants. Choosing xy00 0 and substituting (4.4.5) and (4.2.6) into (4.4.46) and (4.4.47) we have 1(22 2 EuC W)sin \ xA 2|OWJE3 | uCA[ Q ()2()EuC W22 fuA Afu2()cos\ ], ()2()EuC W22 fuA 1(22 2 EuC W)cos \ yA 2|OWJE3 | uCA[ Q ()2()EuC W22 fuA Afu2()sin\ ]. ()2()EuC W22 fuA Denoting 142 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 1 22 2 RA 2|OWJ3 | uCAE , O3 the above relations become 2()fu A cos\ sin \ EuC W xR [], (4.4.48) 2()fuA2 2() fuA2 11 ()EuCW22 () EW uC 2()fu A sin \ cos\ EuC W yR [] . (4.4.49) 2()fuA2 2() fuA2 11 ()EuCW22 () EW uC We mention that R is always real. The quantity under the radical is always positive 1 due to the condition uc2 ! 0 . Using the formulae cos( arctanx ) and 1 x2 x sin( arctanx ) , we denote 1 x2 §·A ' arctan¨¸ 2fu ( ) , ©¹EuC W and obtain xR (sin\'\' cos cos sin ) , yR (cos\'\' cos sin sin ) , that yield (4.4.37) and (4.4.38). From (4.4.5), (4.4.7), (4.4.9) and (4.4.41) we have 222 E E()2EA JC W O3 AC W )ccc \' 2 22 , 22(2AAOEJW3 AuAC) that yields (4.4.42) by integration with respect to s , where 222 33 E()2 E AC J W O3 ACC W W C0 222, 2(2AAuACOEJW32 ) 2(O323Au u ) C1 2 22. (4.4.50) 2OEJW32Au A C The component z of the position vector r is computed from ( 4.2.15 )3 ss zu ds [ uuu ( )cn2 ( wssms ( ), )]d ³³223 3 00 STATICS AND DYNAMICS OF THE THIN ELASTIC ROD 143 ()uu us13E(am(ws ( s ), m ), (4.4.51) 1 w 3 whereEamv((), m) is the elliptic integral of the second kind (4.4.43). We used here the formula v Eamv((),) m mv m cn(,)d2 wm w, (4.4.52) 1 ³ 0 uu23 wheremm1 1 and m . uu13 We see in the virtue of (4.4.37)–(4.4.39) that the rod is confined on a plane when CW 0 and E 0 . In this case the shape of the rod is called the ‘elastica’ by Love. In the general case when CWz0 and E z 0 the shapes of the rod are very complicated and the rod is not confined in a plane, the deformation being spatial. Tsuru has obtained various shapes of the rod analytically and numerically. The case of a simple root and a double root. Let us consider the case C 22W E CW, J O2 . (4.4.53) A 3 This implies that ab , ca and f(u) from (4.4.12) becomes f(u) = au(1)(12 u ) . Therefore, f(u) has a simple root and a double root. We prove the theorem: THEOREM 4.4.3 Given uu12 1, u 3 1 the roots of the cubic equation fu() 0, the Euler angles are uniquely determined from the equilibrium equations (4.4.1)–(4.4.3) ||O ||O us( ) 1 23 sech 2 3 s, (4.4.54) A A WCs4 A ||O \ arctan( tanh( 3 s )) , (4.4.55) 2ACW A W(2ACs ) 4 A ||O M arctan( tanh3 s ) . (4.4.56) 2ACW A Proof The solution of (4.4.8) is expressed in terms of hyperbolic functions. Indeed, consider the solution in the form 2 us( ) A11 sech Bs C1 . Substituting this solution into (4.4.8) and taking account that 2 uABBsc 2sechsinh11 1 s , we obtain by balancing the powers of the function sech z 144 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS ||O ||O C = -1, A 2 3 ,B = 3 ! 0 . 1 1 A 1 A So, the solution of (4.2.8) is ||OO || us( ) 1 233 sech 2 s. A A | O | The minimum of u is –1 at s rf. The maximum of u is ( 12 3 ) as s 0 . A The curve is asymptotically straight along Z-axis. The deformation is localized around s 0 . Using (4.4.31) we have CW \c . (4.4.57) A(1 u ) Substituting (4.4.54) into (4.4.57) we get CW \c . (4.4.58) ||O 2tanhA 2 3 s A Integrating with respect to s we have (4.4.55). Substituting (4.4.53) into (4.4.32) we obtain uCWW() A A Mc , (4.4.59) Au(1 ) or CAWW( C) Mc . (4.4.60) Au(1 ) A Finally, integrating (4.4.60) with respect to s gives (4.4.56). We prove the following theorem THEOREM 4.4.4 Given the conditions of the theorem 4.4.3 the components of the position vector are 2|O3 |Au ( 1) § CsW · x sin ¨ ¸ , (4.4.61) O3 ©¹2A 2|O3 |Au ( 1) § CsW · y cos¨ ¸ , (4.4.62) O3 ©¹2A 4A zs tanh(2 s ) . (4.4.63) O3 Proof Substituting (4.4.53) into (4.4.50) 1 we obtain STATICS AND DYNAMICS OF THE THIN ELASTIC ROD 145 C0 0 . (4.4.64) From (4.4.42) we have CW ) s . (4.4.65) 2A The relations (4.4.61) and (4.4.63) are derived from (4.4.37) and (4.4.38). From (4.4.39) it results (4.4.63). 6 For graphical illustration we consider the values E =194200 u 10 Pa, Q1 =0.28, U = 7876 kg / m3 , l = 10 m, and a = 2 u 103 m. The shape of the rod of infinite length into which the central line is deformed is called elastica. Figure 4.4.3 displays four shapes of elastica for W 0 and different set of values (E =0.3,OJ3 = 0.4, =0.2), ( E =0.7, O3 = 0.2, J =0.1), ( E =0.3, O3 = 0.1,J =0.1) and (E =0.9,OJ3 = 0.4, =0.3) from left to right. These shapes are similar to the shapes of elastica found by Love in 1926. The case Wz0 is illustrated in Figure 4.4.4. We see that for Wz0 the rod deviates from a plane and has a three-dimensional structure. This structure is simpler for small values of WW and more complicated when increases. Figure 4.4.3 Shapes of elastica of Love for W 0 (k = 0). In Figure 4.4.4, the following sets of parameters were considered from left to right (,W 0.2 E =0.3,OJ3 = 0.4, =0.2), ( W 0.3 , E =0.7, O3 = 0.2, J =0.1), ( W 0.4 , E =0.3, OJ3 = 0.1, =0.1) and (W 0.5 , E =0.9, O3 = 0.4, J =0.3). In this case, the shape of the rod consists of a single loop or a series of loops lying altogether in space. 146 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS Figure 4.4.4 Four shapes of elastica for Wz 0 . 4.5 Exact solutions of the motion equations We determine the exact solutions of the motion equations using the equivalence theorem 4.3.1. The motion equations (4.2.54)–(4.2.57) are equivalents to equilibrium equations (4.2.35)–(4.2.37) if 22 A okv12 A,, C o[o kv C s. According to this theorem, we have 2 O Uv (sin T cos \ ,sin T sin \ ,cos T ) (0,0, O3 ) , whereO3 is supposed to be known. The period of the function u is given by (4.4.26) 22S 2()Km , (4.5.1) uu13 k U uu k 13 . (4.5.2) 2()K m The frequency of a cnoidal wave is given by b Z kkuu 2( u ). (4.5.3) a 123 The theorems demonstrated in the static case are valid also in the dynamic case. We supposeOT\,, and M are functions of variables [ s vt . We have 2 O ]d33O(0,0, ) , ] Uv , d3 (sinT\T\T cos ,sin sin ,cos ) and O3 a constant. STATICS AND DYNAMICS OF THE THIN ELASTIC ROD 147 The case of three distinct and real roots. THEOREM 4.5.1 Given uuu32 1 the distinct and real roots of the cubic equationfu() 0 , the motion equations (4.2.54)–(4.2.57) have a unique solution for the Euler angles ||O uu ()cn(()(),) u u2 3 u u [[ m 223 2A 13 3 (4.5.4) 2 uuu223( )cn [ w ( [[3 ), m ], uu2 3 ||O3 with m and wu ()13u , uu13 2A 1 E()Ckv 2 W uu \ {[ 2 3wm( [[ ),23,] 4(Akvw 22 ) 2 11uu3 1 3 3 (4.5.5) 2 E()Ckv 2 W uu23 3[[(wm[3 ), , ]}, 11uu33 W[()]CA k kv221 E(Ckv) W M 21 [ { 2 u Akv224( Akvw )221 u 113 (4.5.6) 2 uu23E()Ckv 2 W uu23 u3[wm ( [[33 ), , ] 3( w ( [[ ), ,m)}, 11uu33 1 u3 where 3(,,x zm ) is the normal elliptic integral of the third kind. THEOREM 4.5.2 In conditions given by the theorem 4.5.1, the components of the position vector are xR sin(\' ) R sin ), (4.5.7) yR cos(\' ) R cos), (4.5.8) ()uu zu [13E[am(w ([[ )), m ] , (4.5.9) 1 w 3 with 1 222222 RA 2|OWJ31 |(kv )uC (k 2vA ) (k 1v ) E , (4.5.10) O3 § A · ' arctan¨ 2fu ( ) ¸ , (4.5.11) ©¹EuC W E ) [CwCm 3(,,) [ , (4.5.12) 2A 01 148 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS where f(u) is given by (4.4.8), and C01 , C are constants, and Eamv((), m) is the elliptic integral of the first kind. Unknowns [J3 ,,vm , , O3 , W and E are determined from initial conditions (4.2.67) and boundary conditions (4.4.9). The case of a simple root and a double root. THEOREM 4.5.3 Given uu12 1, u 3 1 the roots of the cubic equation fu() 0, the Euler angles are uniquely determined from the motion equations (4.2.54)–(4.2.57) ||OO3 2 ||3 u()[ 1 22 sech 2 [, (4.5.13) A kv11 A kv 22 ()Ckv21 W[ 4()Akv ||O3 \ 22arctan[ tanh(2 [)] , (4.5.14) 2(Akv12 ) ( Ckv )W Akv 1 22 W[2ACvk (21 2 k )] [ 4(Akv 1 ) ||O3 M 22arctan[ tanh2 [ ]. (4.5.15) 2(A kv12 ) ()C kvW A kv1 THEOREM 4.5.4 Given the conditions of the theorem 4.5.3 the components of the position vector are 2 2 2|O31 |(Akvu )( 1) §·()CkvW[2 x sin ¨ 2 ¸ , (4.5.16) O31©¹2(Akv ) 2 2 2|O31 |(Akvu )( 1) §·()CkvW[2 y cos¨ 2 ¸ , (4.5.17) O31©¹2(Akv ) 4(Akv 2 ) z [ 1 tanh(2[ ) . (4.5.18) O3 Finally, let us remember that the equation (4.4.8) has the form 1 ufuc2 (), (4.5.19) 2 with 11()CkvW22 2 1E2 fu() [ Ou32 ( J 2 )uO u ( J )]. (4.5.20) Akv 2 332222 1 Akv11Akv Chapter 5 VIBRATIONS OF THIN ELASTIC RODS 5.1 Scope of the chapter The first treatment of the partial differential equations of wave motion was made by D’Alembert in 1750 in connection with the vibrations of thin rods. Daniel Bernoulli, Euler, Riccati, Poisson, Cauchy, Lord Rayleigh (1877) and also Strehlke (1833), Lissajous (1833) and Seebeck (1852) are foremost among those who have advanced knowledge in this problem. The vibrating rod has been a point of research in physics and mathematics, at last centuries up to the present day. Its current importance is associated with the theory of vibrating solitons. In this chapter the vibrations of thin elastic rods are studied. In section 5.2 the small transverse and torsional vibrations of the initial deformed rod are considered. The large relative displacements can occur even when the strains are small. The nonlinear effect caused by the geometrical constraint yields to soliton solutions. The transverse vibrations of a helical rod are presented in section 5.3. The rod vibrates in space by bending and twisting. The vibrations are studied around the strained position of the rod which satisfies the static equilibrium equations given by the theorem 4.2.1. In section 5.4 it is shown that for a special class of media that do not remember the interaction process (DRIP media), the waves admit the solitonic features, but in contrast to solitons, the waves distort as they propagate by an amount that is not altered by the interaction. The only effect of the interaction is to alter the arrival time of their fronts at any point. The vibrations of a heterogeneous string are considered in the last section. Before proceeding, we mention the material we referred in this chapter: Love (1926), Landau and Lifshitz (1968), Lewis (1980), Synge (1981), Seymour and Varley (1982), Tsuru (1986, 1987), Munteanu and Donescu (2002). 5.2 Linear and nonlinear vibrations Consider the transverse vibrations of a rod confined on a plane with no torsional displacements. In this case the Eulerian angles M and \ are zero in the motion equations (4.2.54)–(4.2.57). So, we have Ur Oc 0 , (5.2.1) TTOkA113cc cos TO sin T 0 , (5.2.2) 150 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS OT2 sin 0 . (5.2.3) The unit tangential vector along the rod is given by d3 (sin,0,cos)T T. (5.2.4) Substitution of (5.2.4) into (4.2.66) gives 2 UTsin TUT cos TO1cc 0 , (5.2.5) 2 UTcos TUT sin TO3cc 0 . (5.2.6) Equation (5.2.2) becomes TTOkA113cc cos TO sin T 0 . (5.2.7) ForT H 1 , and neglecting terms of the order H2 , equations (5.2.5)–(5.2.7) can be written as (Tsuru) UH O1cc 0 , (5.2.8) O3cc 0 , (5.2.9) O11 HHkA cc . (5.2.10) Substitution (5.2.10) into (5.2.8) leads to the transverse vibrational motion UH kA1 Hcc H cccc 0 . (5.2.11) A wave solution of this equation is given by H H0 exp[i(ks Z t )] , (5.2.12) with H0 a constant. Substituting (5.2.12) into (5.2.11) we have 4 2 Ak Z 2 . (5.2.13) Ukk1 The group velocity vg of this wave is d(2ZEA kk2 ) k v , E 1 . (5.2.14) g d(1)kkUE23/2 U Let us consider now the torsional motion characterized by T \ 0 (Zachmann). The motion equations (4.2.54)–(4.2.57) yields the known wave equation of motion for small linear vibrations MMkC2 cc 0 , (5.2.15) that admits solutions of the form M M0 exp[i(ks Z t )] , (5.2.16) and the dispersion relation VIBRATIONS OF THIN ELASTIC RODS 151 C P Z k k . (5.2.17) k20U For large displacements, even when the strains are small, the nonlinear terms in the motion equations (4.2.54)–(4.2.57) cannot be neglected. In this case we must consider the geometric constraints expressed by rddc 33,| | 1. Consider firstly the plane motion of the rod, in the absence of torsion ( \ M 0 ). The motion equations are given by (5.2.5)–(5.2.7). We may assume that the functions T(,)s t and O(,)s t are functions of the variable [ s vt . Denoting with prime the differentiation with respect to [ , the motion equations (5.2.5) –(5.2.7) become 22 2 UvvTTccsinU TTOcc cos1c 0 , (5.2.18) 22 2 UvvTTcccosU TTOcc sin3c 0 , (5.2.19) 2 ()TOTOTkv113 A cc cos sin 0 . (5.2.20) Note that the equations (5.2.18) and (5.2.19) can be written under the form 2 Uv (cos)TTOcccc1 0, 2 Uv (sin)TTOcccc3 0. Integrating twice with respect to [ , the above equations are 2 O1 Uv sin TJ[11G , (5.2.21) 2 O3 Uv cosTJ[33G , (5.2.22) where JJ131,,G and G3 are constants. By imposing the boundary conditions Tc 0,r 0 as s orf , t of, (5.2.23) it results J13 J 0 . Inserting (5.2.21) and (5.2.22) into (5.2.20) we have 2 ()AkvTGTGT11cc cos 3 sin 0 . (5.2.24) Defining 22 22 G113 GGsin D, G313 GGcos D, the equation (5.2.24) becomes the known simple pendulum equation d2T Z2 sin T 0 , (5.2.25) d[2 0 where we denoted ToTD, and 152 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 22 2 GG13 Z0 2 . (5.2.26) A kv1 The exact solution of (5.2.25) is expressed in terms of elliptic or hyperbolic functions, depending on the initial conditions. For the conditions (5.2.23), the solution of (5.2.25) is T sin tanh(Z[ ) . (5.2.27) 2 0 From (5.2.27) it results T cos 1 tanh2 ( Z[ ) sech( Z[ ) . (5.2.28) 2 0 0 Thus, with the aid of the formulae 22 sin2tanh()sech()T Z[00 Z[ , cosT sech ( Z[0 ) tanh ( Z[0 ) , (5.2.29) the equations (5.2.21) and (5.2.22) become 2 O10 2 Uv tanh( Z[ )sech( Z[01 ) G , 22 2 O30 U2[sech()tanh()]v Z[ Z[03 G. (5.2.30) When the ends of the bar are fixed, the equation (5.2.1) yields Ux Oc 0, 1 (5.2.31) Uz O3c 0. From (5.2.31) and (5.2.30) we obtain the following solutions 2 x sech[Z0 (svt )] , (5.2.32) Z0 2 zs Ztanh[0 ( svt )] . (5.2.33) Z0 Figure 5.2.1 Shape of the elastica. The solution (5.2.32) represents a soliton. The solution (5.2.33) is a kink and represents a twist in the variable z . The shape of elastica, into which the central line is VIBRATIONS OF THIN ELASTIC RODS 153 deformed, is a single loop and it is represented in Figure 5.2.1 ( Z0 8 ). The shape of the kink (5.2.33) is represented in Figure 5.2.2. Since the waves can move in the rod in both directions, a head-on collision is possible. Konno and Ito have studied nonlinear interactions between solitons for KdV and Boussinesq equations. In 1962, Perring and Skyrme investigate the head-on collision of two kinks with equal but opposite velocity. The kink traveling to the left is called an anti-kink. Using the same definition, an anti-soliton is a soliton traveling to the left. It is interesting to investigate the behavior of waves during their mutual interaction. To understand the effect of the collision between waves into the motion of the rod, let us consider next two initial conditions for the equations (5.2.31) 2 x10 sech[Z (svt1 )] , Z0 2 zs10 Ztanh[ ( svt1 )] , (5.2.34) Z0 2 xs202 sech[Z (Gvt )] , Z0 2 zs 20 'tanh[ Z (svt2 G )] , Z0 that represent two waves traveling with the velocities vv1 ! 2 , in opposite directions. Figure 5.2.2 Shape of the kink solution. The result of integrating the equation (5.2.34) 1 is shown in Figure 5.2.3, and the result of integrating the equation (5.2.34)21 in Figure 5.2.4, for v 0.5 , v2 0.35 , ' 0.75, G 1.75 and Z0 2.44 . The two solitons are separated before the collision, then coalesce in the interaction zone and separate again afterwards, without change of velocity and shape, but with a small phase shift. In the collision zone there is no linear superposition. In contrast to the elastic soliton collision, the collision of kinks is inelastic because after collision, a small amount of energy is left in the form of oscillations. 154 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS The two kinks keep in some way their structure, but the oscillations indicate a dissipation of energy. After collision the velocities of kinks are v1c 0.46 andv2c 0.33 . The coalescing of two kinks in the interaction zone (,)s t is represented in Figure 5.2.5 for two cases: v1 0.5 ,v2 0.35 , and vv12 1. In the second case, after collision the kinks velocities are vv12cc 0.87 . Figure 5.2.3 Collision of a soliton and an anti-soliton traveling in opposite directions with different velocities. The solitons and kinks behave stably in the collision process even if the interaction between them takes place in a nonlinear way, where we cannot apply the principle of the linear superposition to the process. In the interaction region the kinks behave in a complicated way. A projection of the collision zone on the plane (,)s t can be seen in Figure 5.2.5. VIBRATIONS OF THIN ELASTIC RODS 155 Figure 5.2.4 Collision of a kink and anti-kink traveling in opposite directions with different velocities. Figure 5.2.5 Projection of the interaction zone on the plane (,)s t of two kinks of velocities v1 0.5 , v2 0.35 (left) and vv12 1 (right). 5.3 Transverse vibrations of the helical rod Let us consider a rod that vibrates around the strained position, which satisfies the static equilibrium equations given by the theorem 4.2.1 for O (0,0, O ) . We begin with differentiating the equation (4.2.54) with respect to s (Tsuru) Ud Occ 0 . (5.3.1) 3 Suppose that the strained rod has a helical form. The Euler angles are written as T T0 (s ) H cos(kx Z t ) , 156 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS \ \0 (s ) H sin(ks Z t ) , (5.3.2) M M0 (s ) H sin(kx Z t ) , where T\000(),s (),s M ()s are solutions of the equations (4.2.54)–(4.2.57), and H a small parameter. Substituting (5.3.2) into (5.3.1) we obtain an equation in O Occ Ud3 , (5.3.3) with 2 d3 ( T cos T cos \ 2 T\ cos T sin \T sin T cos \ \T\\T\TT\T\T\ sin sin 2 sin cos , cos sin 2 cos cos (5.3.4) TT\\T22sin sin sin cos \\T\TTTT sin sin , sin2 cos ). Then, differentiate with respect to s the equations (4.2.55)–(4.2.57) and insert Occ given by (5.3.3). After little manipulation with neglecting terms of third order in H , we obtain the vibrations equation written in a matrix form M H 0 , (5.3.5) where M is a 3u3 symmetric matrix of elements 2 4222 Z 2 (3)cos\cc000k \ T Mkk11 2222{}U 1 kk()\c0 (5.3.6) 22 2 {(CA 2 )cos T\W\T0000 ( AC )}c C cos Ak , 2kUZ2 MM12 21 222 \TTc0cos 0 sin 0 ()k \c0 (5.3.7) TkCACsin00 {( 2 )cos T\Wc0 , 22 MM13 31 (2 k 1Z Ck )cosT 0 , (5.3.8) 22U 222 Mkk22 Z{()sin(cos222\c0 T0 1 T 0 1)} ()k \c0 (5.3.9) 22 2 TTkC( cos00 A sin ), 22 MM23 32 (2 k 1ZT Ck )cos 0 , (5.3.10) 2 2 M 33 2kCk 1Z , (5.3.11) andH a column vector H (,,) HHHt . (5.3.12) In (5.3.6)–(5.3.11) we have used the relations kk21 2 , Mcc00 W\cos T0 . (5.3.13) VIBRATIONS OF THIN ELASTIC RODS 157 The dispersion relations are calculated from detM 0 , (5.3.14) where the determinant of M is a cubic polynomial of Z2 . Consider the case of transverse vibrations. The characteristic equation is given by (Tsuru) 2222 422222 2 U[(1ukkkAukku00 )\cc (1 1 )] ZU [ 000 \ ( \c 13 10 ( 1)) 22 4222 2 W\Cu00cc(5)2( \ 0 k Ak k \c 0 kk 1 2] ku 10 Z 42 2222 2 2 22 \kkAuACuAkkuC()[3cccc00 \W\\00001 ( 2)]0 W 0, (5.3.15) where u00 cosTzr 1. The equation (5.3.15) is a cubic polynomial in Z2 . We note with (Z ) 2 and Z )( 2 the roots of (5.3.15) which are functions of k. Numerical investigations show that for 22 2 2 22 2 c0 \ k , the roots Z )( and Z )( are positive, and for \c0 t k the root (Z ) is positive and Z )( 2 is negative. The waves are stable for real values of the angular frequency Z . Next, we consider 22 only the case \c0 k . The initial strain is determined by the parameters u0 , \c0 and W . We represent graphically the variation of Z )( 2 and (Z ) 2 with respect to k, for different values of 2 u0 ,\c0 andW . These are the dispersion curves, calculated for A =2.4404 Nm , C =1.9066 Nm2 and U =7876 kg/ m3. Let us choose u00=0.3 and \c [0,1,2,3] . Figure 5.3.1 corresponds toW 0 , and Figure 5.3.2, toW 0.5 . Then we take W[0,1,2,3] and \c0 1 . Figure 5.3.3 corresponds to u0 =0.3, and Figure 5.3.4 to u0 0.7 . From these figures we observe that the dispersion relations depend strongly on all parameters. Figure 5.3.1 Dispersion relations of the transverse vibrations of a helical rod for different values of \c0 . 158 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS Figure 5.3.2 Dispersion relations of the transverse vibrations of a helical rod for different values of \c0 . Figure 5.3.3 Dispersion relations of the transverse vibrations of a helical rod for different values of W Figure 5.3.4 Dispersion relations of the transverse vibrations of a helical rod for different values of W VIBRATIONS OF THIN ELASTIC RODS 159 Finally, we represent in Figs 5.3.5 and 5.3.6 the shape of the vibrating rod at t 35 and t 124 , for \c0 =2, W =0.5, u0 =0.2, O3 =0.4. The initial value of Mc0 is calculated from (5.3.13). Figure 5.3.5 Shape of the vibrating rod at t 35 . Figure 5.3.6 Shape of the vibrating rod at t 124 . 5.4 A special class of DRIP media Fermi, Pasta and Ulam studied in 1955 the oscillations of a heterogeneous string which is governed by nonlinear wave equations of the form 2 yAyyytt (,)x t xx , (5.4.1) dA A3/2(PQA) , (5.4.2) dyx whereyx(,)t is the physical displacement, A(,)yyx t a positive function representing the local speed of propagation, and PQ, the material constants, and x ranges from f tof . In particular, when A Ax(), the equation (5.4.1) is applied to all physical systems essentially involving only one space-dimension and onetime-dimension. For the transverse vibrations of a string, we have A2 Tm/ where T is the constant tension and m the mass per unit length function of x , for the compressional vibrations of an isotropic laminated elastic solid in which the density and elastic constants are functions of x only, A2 (2OP)/U , and for the transverse vibrations of a laminated solid, A2 P/ U. In 1982 Seymour and Varley studied the equation (5.4.1) and have shown that, for certain functions A()yx that satisfies (5.4.2), the solutions of (5.4.1) are simple waves 160 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS whose profiles can be specified arbitrarily, that have the properties: they interact between themselves like solitons, being unaffected by the interaction, but in contrast to solitons, distort as they propagate by an amount that is not affected by the interaction. Since the interaction phenomenon for waves of arbitrary shape and amplitude is a property of the transmitting medium rather than of the particular wave profiles, Seymour and Varley named these media DRIP media (media that do not remember the interaction process). In such media the profile of the interacting waves is not affected by such interactions. The equation (5.4.1) can be written as a system of first order equations 2 uAeuet (, )x , uex t , (5.4.3) or under the form uAuAeAetx () tx , uAuAeAetx ( tx ) , (5.4.4) whereuy t and ey x . The problem of integrating (5.4.3) may be reduced to the determining of the functions xx (,)DE and tt (,)DE that satisfy the equations xEE At , xDD At , (5.4.5) where(,DE) are characteristic parameters (,DE) . When A depends only on e , integration of (5.4.6) yields e uG F()ED ( ), ()()ce() GFED, ce() ³ As ()d s. (5.4.6) 0 Consequently, (5.4.5) must be regarded as xEE Act() , xDD Act() , (5.4.7) where A is a function of cG F()ED ( ). The equation (5.4.4) becomes uAeEE , uAeDD . (5.4.8) The equations are very difficult to be integrated, but there are some exceptions when either G 0 or F 0 . For G 0 , relations (5.4.6) and (5.4.7) yields cu F() D , xA() Ft D, (5.4.9) that describe a right-traveling simple wave. For F 0 , relations (5.4.6) and (5.4.7) imply that cu G() E, xA() Gt E, (5.4.10) that describe a left-traveling simple wave. Both solutions represent waves moving with the velocity A()c into a uniform region where uc and are constant. Let us consider now the class of DRIP media, that include a large variety of elastic– plastic materials, composite materials, gases, foams, and so on. VIBRATIONS OF THIN ELASTIC RODS 161 DEFINITION 5.4.1 (Seymour and Varley) A DRIP medium is a nondispersive medium that transmits waves that do not remember the interaction process. The waves are of arbitrary shape and amplitude and distort in time but by an amount that is not affected by the interactions. A DRIP medium is defined by the condition that A()c satisfies an equation of state of the form dA PA1/2 QA 3/2 , (5.4.11) dc whereP ,Q are material constants. The equation of state (5.4.11) contains two arbitrary parameters that is an advantage to be used for a model of a wide class of different media. The interaction of this type occurs when two solitons collide. The difference is that solitons are represented by waves of permanent form whose profiles are specific. We will show that for a DRIP medium the equations (5.4.7) can be integrated to obtain a simple representation for x(,)DE and t(,)DE . THEOREM 5.4.1 (Seymour and Varley) The general solutions of (5.4.7) are given by xlrARL(,)DE [() E ()] D1/2 P (() D ()) E , tl(,)DE [() E rA ( D )]1/2 Q ( R ( D ) L ()) E , (5.4.12) where LlGcc()E ()E, RrFcc()E (D ), (5.4.13) andA(,)DE is determined from cG ()ED F ( )and (5.4.11), and the functions F,,,,,GRLrl are determined from initial and boundary conditions. Proof. The proof of this theorem belongs to Seymour and Varley. Return to (5.4.7) and eliminate x . Thus, t(,)DE will satisfy the equation ww ()At ()0 At , (5.4.14) wDED wE or, if we take into consideration that cG ()ED F ( ) w (2At1/2 )E ( AcAtGcc ( )1/2 ) ( ) 0 . (5.4.15) wD ED This equation can be easily integrated with respect to D . Using (5.4.11), integration of (5.4.15) yields 1/2 2()()AtE PQ t xGcc E g() E, (5.4.16) where g is an unspecified function. From (5.4.16) and (5.4.7) 1 we obtain wPªºA1/2 1 QA «»()PtxQ gc () E 0 . (5.4.17) wE¬¼ P QAA2 P Q 162 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS In a similar way wPªºAA1/2 1 Q «»()PtxQ fc () D 0 , (5.4.18) wD¬¼ P QAA2 P Q wheref is an unspecified function. Relations (5.4.17) and (5.4.18) leads to 1 P QA wI 1 P QA wI PQtx , PtxQ , (5.4.19) fAc()DwD1/2 gAc()E 1/2 wE where wI2 1 PQA()c Z()cfcc ( D ) g ()E , Z()c . (5.4.20) wDwE 2 PQA()c On the basis of (5.4.11) it results that the function Z()c defined above verifies the equation ZPQZcc()cc () 0. (5.4.21) 1 wI 1 wI Denoting q and p , we obtain from (5.4.20) f c()DwD gc()E wE wq 1 wp 1 Z()cgc ( G ), Z()cfc ( F ), (5.4.22) wG 2 wF 2 where it is convenient to regard (,qp ) as functions of (,F G) . Integrating (5.4.22) we obtain the solutions qLGc cc()()ZZ LGc () () rF (), pRFc cc()()ZZ RFc () () lG (), (5.4.23) where 1 1 LLgcc PQ c , RRfcc PQ c . (5.4.24) 2 2 In order that (,)x t given by (5.4.19) and (5.4.23) to satisfy the equations (5.4.7) it results that the functions l and r must be given by lG() Lc () G , rF() Rc () F . (5.4.25) From (5.4.19), (5.4.20) and (5.4.23) it follows the solutions (5.4.12). The relation (5.4.13) is obtained from (5.4.25). When A is independent of c , A AL , we see that (5.4.7) can be integrated xAtL D, xAtL E, (5.4.26) and, therefore, u and c are given by VIBRATIONS OF THIN ELASTIC RODS 163 uGxAtFxAt ()(L L ) , cGxAtFxAt ()(L L ) . (5.4.27) These representations of the linear theory are interpreted as the superposition of two nondistorting and noninteracting waves that move in opposite directions with speed AL . P These formulae can be obtained from (5.4.12) and (5.4.13) by taking A()cA . Q L Now, we attach to (5.4.7) the initial conditions ux(,0) u0 () x, cx(,0) cx0 (), f x f. (5.4.28) Normalizing (,)DE so that at t 0 to have D x and E x , it follows from (5.4.4) 1 1 F()xcxux [ () ()], Gx() [ c () x u ()] x . (5.4.29) 2 00 2 00 From (5.4.12) we obtain the solutions 1/2 x [lx ( ) rx ( )] A0 ( x ) P ( Rx ( ) Lx ( )) , 1/2 0 [()lx rx ()] A0 () x Q ( Rx () Lx ()), (5.4.30) where A0 ()x is determined from cx0 () by (5.4.11). The equations (5.4.30) and (5.4.13) and (5.4.29) determine the functions RLrl,,, . 5.5 Interaction of waves We present the Seymour–Varley method for analysis of the interaction of any two waves that are traveling in opposite directions in DRIP media. The waves meet and interact and then emerge from the interaction region unchanged by the interaction. Following this method, we consider an interaction region in the plane (,)x t composed of five regions in which u z 0 and c z 0 , as shown in Figure 5.5.1. At t 0 the right- traveling wave (RW) occupies the region xlfddxx,xxlf, ! 0 , and the left- traveling wave (LW) the region 0 ddx f xxr . The waves move into a region where uc 0 . Suppose u0 ()x andc0 ()x are continuous. Suppose that at t 0 we have F()xf ()x for dxlfx dx , and zero otherwise, and Gx() gx () for x frddxx, and zero otherwise. In region I0R we have G , F z 0 , L 0 ; in I0L , F , G z 0 , R 0 ; in region II , F z 0 , G z 0 ; in regionIIIR , G 0 , F z 0 , L LR and in regionIIIL , F 0 , G z 0 , RR L . We have also fx()lffr fx ( ) gxgx ()() 0(. The function A c) is supposed to be defined as 164 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS A(fx ( )), for dd xlf x x, ° A(cAxAgx ) ( ) ® ( ( )), for xxxfrdd, (5.5.1) ° ¯ AA(0) 0 , otherwise. We also see that for ||x d x f we have x RL 0 , lr 1/2 , (5.5.2) 2A0 and for ddxlfxx I()xIx ( ) L 0 , RA 1/2 () x f , PQA()x 1 x lIxIx [( ) () ], 2 f A1/2 ()x (())()P QA xRx x r , (5.5.3) 2()Ax1/2 where x x ds Ix() . (5.5.4) 1/2 ³ 1/2 A ()xAs0 () Also, we have for x frddxx I()xIx () R 0 , LA 1/2 () x f , PQA()x 1 x rIxIx [() ( ) ], 2 f A1/2 ()x (())()P QA xLx x l , (5.5.5) 2()Ax1/2 and for x t xr 1/2 I()xIxf ()r R 0 , LL r A0 , (5.5.6) PQA0 ()PQAL00rr ()PQAL x lr1/2 1/2 1/2 . (5.5.7) 22A00AA2 0 VIBRATIONS OF THIN ELASTIC RODS 165 Figure 5.5.1 The plane (x ,t ) of the wave interaction In the region IR , the RW is a simple wave, in the region II it interacts with the LW, and in the region IIIR it is again a simple wave. In a similar way, in the region IL , the LW is a simple wave, in the region II it interacts with the RW, and in the region IIIL it is again a simple wave. The important role of this analysis consists in that a simple wave in IIIRR is the same as the wave that is in I m and that the simple wave in IIIL is the same as that in IL . To show this, let us analyze the RW. Since G 0 in IR andIIIR , the relation (5.5.1) gives AA ()D, dDdxlfx . (5.5.8) In the region IR , the solutions (5.4.12) become xl [(ED ) r ( )] A1/2 PD R ( ), tl [(ED ) r ( )] A1/2 QD R ( ). (5.5.9) Eliminating l()E in the relations (5.5.9) we have xAD() t D. (5.5.10) Mention that in the relation (5.5.10), A is given by (5.5.8), l()E by (5.5.2) and Rr(),()DD by (5.5.3). In the region IIIR , Rr (DD ), ( ) are given by (5.5.3), and l()E and LL()E r are given by (5.5.7). Thus, here the solutions (5.4.12) are identical with (5.5.9) if x and t are replaced by x xLPr , tx PLr . (5.5.11) Therefore, in the region IIIR we have a RW given by xA D() t D. (5.5.12) So, by comparing (5.5.1) and (5.5.12) we see that the emerging wave is not affected by interaction, it being identical with waves produced by the initial conditions tL Q r , 166 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS F()xfxL (Pr ) for PddPxlrLx x f L r, and zero otherwise, and Gx() 0 for all x . The effect of interaction is only to change the effective origin of x and t in the x x original wave. If the RW is before the interaction, a centered wave with A f , t x PLx then after interaction, the emerging wave has A rf. tLQ r To illustrate this, we consider a simple wave profile A()x sech x . This is the case of interaction of two pulses having a soliton profile, traveling in a DRIP medium. The calculations are performed numerically and presented in Figure 5.5.2. In Figure 5.5.2 we see that, in contrast to known theory of solitons interaction, these pulses travel in opposite directions, interact and emerge unaffected by the interaction. In the interaction region no coupling between waves is visible. This suggests that the waves may be regarded individually. Speaking from a physical viewpoint, this interaction requires that the energy of each field is carried individually without any transfer of energy between fields. This property may be of the transmitting medium rather than of the particular wave profiles. Figure 5.5.2 Profiles of waves against x for several t 5.6 Vibrations of a heterogeneous string Synge studied in 1981 the vibrations of a heterogeneous string and has shown that the solutions can be obtained by applying a linear integral operator to a Cauchy function of position and time furnished by initial conditions. In the following we present the Synge method applied to the partial differential equation 2 yAxtt ()yxx , (5.6.1) whereA()x is a positive function. Synge method straighten the characteristics of (5.6.1) by using the transformation VIBRATIONS OF THIN ELASTIC RODS 167 x dz xuxo() ³ , (5.6.2) 0 A()z that yields c yyu y 0 , (5.6.3) tt uuc u wherecux(()) Ax () is transformed local speed. In the Cartesian coordinates (,)ut , the characteristics are straight lines inclined to the axis at 45o . By a change of variable yut(,) vut (,)()I u , (5.6.4) the equation (5.6.3) becomes, on division by I vvtt uu 22 kvhv u , (5.6.5) with I c IIc 22k uu, 22h uu u u . (5.6.6) I c IIc For I c , we have k 0 , and equation (5.6.5) reduces to vvtt uu 2 hv. (5.6.7) Integrating (5.6.7) over the triangle PAB (Figure 5.6.1) where Put(,) is a generic point, we have (2)dd(dd)2ddv v hv u t v t v u hv u t 0 , ³³ uu tt ³u t ³³ or 1 vP() [() vA vB () v dd] u t hvu dd t. (5.6.8) ³³³t 2 AB Figure 5.6.1 The triangle PAB in the plane (,)ut . 168 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS Introduce now the Cauchy function determined by the initial conditions for v and vt for t 0 1 u1 Cut( , ) [ vu ( t ,0) vu ( t ,0) v ( z )dz] , (5.6.9) ³ t 2 u1 which satisfies the equation CCtt uu 0 . (5.6.10) Then introduce a linear integral operator H to be applied to function vu(,)t Hvut(,) hu ( )( vu , t )dd u t . (5.6.11) ³³ 11111 With (5.6.9) and (5.6.11), we obtain from (5.6.8) the integral equation for the solution vut(,) vut(,) Cut (,) Hvut (,). (5.6.12) To solve (5.6.12) we write it as Cut(,) (1 Hvut )(,), and then multiply it by (1 H ) 1 , and obtain an infinite sequence of operations applied to a specified Cauchy function, that is vut( , ) (1 H )12 C (1 H H ...) C. (5.6.13) The series converges if C and h are bounded in absolute values. This solves completely the problem of vibrations of a heterogeneous string. Next, we present the Lewis method to solve the equation (5.6.1) for the initial data yx(,0) f () x, yxt (.0) gx (). (5.6.14) The energy E of the string is given by 1 f yy22 E ³ utdu . (5.6.15) 2()f cu Since the energy is conserved, the flow on phase-space, induced by the propagation of a wave according to (5.6.3) conserves the inner product 11 !vv12,(, cvcv 1 2) , §·cx() 0 c ¨ ¸ , (5.6.16) ¨ ¸ ©¹0(cx) where 1 f (vv , ) [ ququ () () pupu () ()]d u. 12 ³ 1 2 1 2 2 f Here,v is a point in the phase-space (,yyut) VIBRATIONS OF THIN ELASTIC RODS 169 §·q v ¨¸, qu() yuu (,0), pu() yt (,0) u . (5.6.17) ©¹p Therefore, from (6.4.3) we have the equation governing the flow d vt() Lvt (), (5.6.18) dt where §·yu §·0 wu cu v ¨¸, L ¨¸, 2()J u . (5.6.19) ©¹yt ©¹wJu 20 c We calculate vt() by a sequence of linear transformations that reduce vt() to a perturbation of the pulse. The operator L is screw-symmetric with respect to the inner product (5.6.16) and so there exists a one-parameter group Vt() of orthogonal transformation determined by d Vt() LVt (), V (0) 1, (5.6.20) dt so that vt() Vtv () is a solution of (5.6.18). d We based the following on the fact that exp(tL ) L exp( tL ) , where dt LL23 L exp LEL ... lim (E )m , 2! 3! mof m with E the unit matrix, and det(expLL ) exp(tr ) . Since trL 0 , we have det(exp L ) 1. Also we have exp(wtx ) Tt ( ) , with Tt() the right translation by t [()]()Tt f x f ( x t ), and exp()txw Tc () t, with Ttc() T ( t ) fx ( t ), the left translation by t . The method is based on the decomposition of the phase-space (,yyut) into a pair of complementary subspaces. This induces a decomposition of each initial datum into a forward-propagating part and a backward-propagating part. c In the homogeneous case (0)J u , the equation (5.6.3) is reduced to 2c yytt uu 0 and the solution is expressed as a sum of two waves fx( t) and fx() t that propagate independently. In the heterogeneous case both pulses are coupled by Jz0 considered as a perturbation. So, we take Vt() cRVtRc11 () , Vt( ) exp(tL ) , (5.6.21) 170 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS LcRLRc 11 , u §·w J L ¨¸, ©¹J wu §·1 2 ¨¸2 R ¨ ¸ . (5.6.22) ¨¸1 ¨¸2 ©¹2 It results Vt() Rc11 V () t cR , LRcLcR 1 1 . (5.6.23) We see that L can be written as a sum of two operators to separate the contribution of the coupling term Jz0 u 0 0 §·w § J· LL 0 *, L0 ¨¸, * ¨ ¸ . (5.6.24) ©¹0 wu ©¹J 0 In the homogeneous case we have * 0 and Tt() 0 § · V ()tUtUt (), () ¨ ¸ , (5.6.25) ©¹0(Ttc ) where Tt() is right translation by t [()]()Tt f s f ( s t ), (5.6.26) and Ttc() Tt ( )is left translation by t . The initial conditions (5.6.14) can be written under the form yu(,0) M () u , yut (,0) \ () u. (5.6.27) For* 0 the solution of yytt uu 0 is written as D’Alembert formula 11ut yut(,) [(MM u t ) ( u t )]³ \ ()d z z. (5.6.28) 22ut Our aim is to obtain a similar formula for the inhomogeneous case *z0 . For this we use the well-known perturbation formula (Kato) t Vt() Ut ()³ Ut ( * s ) Vs ()d s. (5.6.29) 0 From this we can obtain an infinite series for Vt() by an iteration scheme t V(1)nn () t Ut ()³ Ut ( * s ) V() ()d s s, 0 VIBRATIONS OF THIN ELASTIC RODS 171 VtUt(0) () (). (5.6.30) We take into account that Vt( ) exp(tL ) maps forward-going data into forward- going data and backward-going data. So, we write §·VtVt() () Vt() FF FB . (5.6.31) ¨ ¸ ©¹VtVtBF() BB () Here, VFF maps forward-going data into forward-going data, VFB maps forward- going data into backward-going data, VBF maps backward-going data into forward- going data and VBB maps backward-going data into backward-going data. From (5.6.25) and (5.6.30) we obtain for VFF t t1 V () t Tt () Tt ( JJ t ) Tc ( t t ) Tt ( )dd t t ... . (5.6.32) FF ³³ 112212 00 The first term in (5.6.32) is simply translating a forward-going datum into a forward direction. The integrant Tt()()(JtTttTt112c J 2 ) translates a forward-going datum in the forward direction from time zero to time t2 when it is reflected. On reflection it is multiplied by the local reflection coefficient J , then translated backwards from time t2 to time t1 , when it is reflected again, multiplied by J and translated forwards from time tt1 to time . So, the second term represents the contribution to the forward-going disturbance from all possible double reflections. The following terms consider third reflections and so on. Knowing this, it is easy to write t Vt () TttTttc (J ) ( )d FB ³ 111 0 (5.6.33) t tt12 JJJTtcc()()()()ddd... t Tt t Tt t Tt tt t , ³³³ 112233123 000 t Vt () TttTttc ( ) J ( )d BF ³ 111 0 (5.6.34) t tt12 JJJTt()()()()ddd... t Tcc t t Tt t T t t t t , ³³³ 112233123 000 t t1 Vt () Ttcc () TttTttTttt ( JJ ) ( ) c ( )dd ... . (5.6.35) BB ³³ 112212 00 Now, from (5.6.31) and (5.6.32)–(5.6.35) we have 172 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS §Vt11() Vt 12 ()· Vt() ¨ ¸ , (5.6.36) ©¹Vt21() Vt 22 () with 11 Vt11 ()[()()()()] cVtVtVtVtFF BB FB BF , 2 c 11 Vt12 ()[()()()()] cVtVtVtVtBB FF FB BF , 2 c 11 Vt21 () cVtVtVtVt [BB () FF () BF () FB ()] , (5.6.37) 2 c 11 Vt22 () cVtVtVtVt [FF () BB () FB () BF ()] . 2 c Taking account of the initial data (5.6.27) we have 11 yuttB( ,) cuV ( )[B () t VFF () t VBF () t VFB ()] tMc ( u ) 2 cu() (5.6.38) 11 \cuVtVtVtVt()[BB () FF () BF () FB ()] (), u 2 cu() and 11t yut( ,) M ( u ) cuVtVtVtVt ( ) [BB () FF () BF () FB ()] Mc ( ut )d1 2 ³ cu() 0 (5.6.39) 11t \cuVtVtVtVt() [ () () () ()] ()d. ut ³ BB FF BF FB 1 2 0 cu() When cuc() 0, (5.6.39) becomes 11tt yut(,) M () u [ Ttcc () Tt ()]()d M u t [ Tt c () Tt ()]()d \ u t. (5.6.40) ³³11 1 11 1 2200 After integration by parts we obtain D’Alembert formula. Chapter 6 THE COUPLED PENDULUM 6.1 Scope of the chapter In the first four sections, the inverse scattering transform is applied to solve the nonlinear equations that govern the motion of two pendulums coupled by an elastic spring. The theta-function representation of the solutions is describable as a linear superposition of Jacobi elliptic functions (cnoidal vibrations) and additional terms, which include nonlinear interactions among the vibrations. Comparisons between the cnoidal and LEM solutions are performed. Finally, an interesting phenomenon is put into evidence with consequences for dynamics of the coupled pendula. The reason why a pendulum is chosen is that its dynamics is rich and complex and its equations are strongly related to the Weierstrass equation with a polynomial of higher order, that admits an analytical solution represented by a sum of a linear and a nonlinear superposition of cnoidal vibrations. The modal interactions and the modal trading of energy were studied in the early 1950s by Fermi, Pasta and Ulam. In the section 6.5 we present the results of Davies and Moon concerning the modal interaction, characterized by highly localized waves, in an experimental structure. These waves are similar to Toda solitons. The extension of the Toda interacting equation is given by Toda–Yoneyama equation, which is presented in the last section. This chapter is refers to the works by Donescu (2000, 2003), Munteanu and Donescu (2002), Davies and Moon (2001) and Yoneyama (1986). 6.2 Motion equations. Problem E1 Figure 6.2.1 shows a coupled pendulum consisting of two straight rods OQ11 , OQ22 of masses M1 , M 2 , lengths OQ11 OQ 2 2 a, and mass centers C12,C with OC11 l 1 , OC22 l 2 and OO12 l . The rods are linked together by an elastic spring QQ1 2 , Q1 OC11,Q2 OC22 characterized by an elastic constant k . The elastic force in the spring is given by kO| 12O Q1Q2 | . The kinetic energy T of the system is 1 TII ( TT22) , (6.2.1) 2 11 2 2 174 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS whereT1 and T2 are the displacement angles in rapport to the verticals, I1 is the mass moment of inertia of OO12 with respect to C12 and I is the mass moment of inertia of OO34with respect to C2 . The elastic potential is written as (Teodorescu) k U gMl(cosTT Ml cos)( OO QQ )2 , (6.2.2) 11 1 2 2 22 1 2 1 2 where QQ22 [ OO a (sin T sin T )] a2 (cos T cos T )2 12 12 2 1 2 1 (6.2.3) 22 OO122 aOO 12 (sin TTTT 2 sin 1 ) 2a [1 cos(2 1 )]. The generalized force is wOC11 A QF1 cos ZTt ( sin1 i cos T 11111 jl )( sin T il cos T j ) , wT11l whereij, are unit normal vectors and QA1 cos Z t. (6.2.4) From Lagrange equations, where LTU d §·wwLL Q , ¨¸ 1 dt ©¹wT1 wT1 (6.2.5) d §·wwLL 0, ¨¸ dt ©¹wT2 wT2 we derive the motion equations of the pendulum k w 2 I Mglsin (OO QQ ) A cos t , ° 11T 1 1 T 1 1 2 1 2 Z ° 2 wT1 ® (6.2.6) k w °IMglTsin T ( OOQQ )2 0, ° 22 2 2 2 1 2 1 2 ¯ 2 wT2 with g the gravitational acceleration. Equations (6.2.6) are coupled and nonlinear. Substitution (6.2.3) into (6.2.6) gives 2 °I11TMgl 1 1sin T 1 kH [ al cos T1 a sin( TT2 1 )] A cos Z t , ® (6.2.7) 2 ¯°IMglkHala22T 2 2sin T 2 [ cos T2 sin( TT2 1 )] 0, where l <(, T12 T ) +T(,12 T ) , (6.2.8) 221/2 Figure 6.2.1 The coupled pendulum. Defining the dimensionless variable k W tt Z1 , (6.2.10) M1 and notations Mgl2 MMgl < 11 ww,, 122 E) , Ik Ik l 12 (6.2.11) a AM alM alM [,,111 G D , D , lkII11 I 2 equations (6.2.7) are reduced to the dimensionless equations °T1112121wsin TJ (TTD , ) [cos T[ sin(TT )] G cos ZW , (6.2.12) ® ¯°TE221wsin TJTT ( ,2221 ) D [cos T[ sin( TT )] 0, M where Z Z 1 , and the dot means the differentiation with respect to W , and k 1/2 J(,TT12 ) ) 1, 2 )T(,12 T )12(sin [ T2 sin)2(1cos( T 1 [ TT2 1 )). (6.2.13) The parameters M 2 and l are supposed to be specified, whereas m , [ , k and G are considered the control variable parameters, where M 1 m . (6.2.14) M 2 The imposed initial conditions are 0000 T1122112(0) TT , (0) TT , (0) TTp , (0) Tp2 . (6.2.15) Noting 176 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS T zzz,,, T T T z , 11 2213 24 (6.2.16) cosZW zz56 , Z sin ZW , the equations (6.2.12) become zz13 , zz24 , (6.2.17) zwzzzz311 sin J ( ,2121 ) D [cos [ sin( zzz )] G 5 , zwzzzz421 Esin J ( ,222 ) D [cos [ sin( zz 1 )], zz56 , 2 zz65 Z , with initial conditions (6.2.15) zzzzzzz(0) 000 , (0) , (0) , (0) z0 , 1122334 4 (6.2.18) zz56(0) 1, (0) 0. S Consider now the case ||z d , and let us approximate the trigonometric functions p 2 by series expansions (Abramowitz and Stegun) zz35 sin zz H(), z | H ()| z d 2104 , 3! 5! (6.2.19) zz24 coszzz 1 H (),| H ()|910. d 4 2! 4! Substitution of (6.2.19) into (6.2.17) yields to a system of equations we refer to as the problem E1 zz13 , °zz , ° 24 ° °zwPzQzzzzz311 () D (,121 )(, J2 ) G 5 , (6.2.20) ®zwPzQzzzz E() D (,)(,), J ° 4221212 °zz , ° 56 2 ¯°zz65 Z , with 35 Pz() z az bz, Qz112(,zRzRzz ) 11 () 212 (, ), 2 4 Qz212(,zRzRzz ) 12 () 212 (, ),,Rz1 () 1 czdz 1/2 J(,zz12 ) 1 f (, zz12 ), THE COUPLED PENDULUM 177 32 Rzz212(, ) [ z 1 [ z 2 az [ 2 3 azz [ 21 235 4 [35azz21 [[ az 1 bz 2 bzz [ 21 32 23 4 5 [10bzz21 [ 10 bzz 21 [[ 5 bzz 21 bz 1, 222 fzz(,12 ) 12[ z1 [ 2 z 2 4 c [ zz 12 2 c [ z 1 22 3 3 24 24 23 [222228cz21 az [ az [ 2 dz [ 2 dz [ 1 dzz [ 21 (6.2.21) 222 2 3 5 5 [12dzz21 [ 8 dzz21 [[ 2 bz 2 2 bz 1 , 1 1 1 1 with a , b , c , d . 3! 5! 2! 4! 6.3 Problem E2 A way for analyzing a system of equations by following LEM procedure given in section 1.6 is to simplify the exact system of equations (6.2.20) by assuming 2 JTT(,12 ) [ (sin T2 sin) T 1 [ (1cos( TT2 1 ), (6.3.1) under conditions 2 |2[TT[TT (sin21 sin ) 2 (1 cos(21 ) | 1 . (6.3.2) Substitution of the function J(,TT12 )given by (6.3.1) into (6.2.17) yields zz13 , zz24 , 23 zwz31 sin D[ cos z 1 D[ sin( zz 211 ) D[ cos zz sin 2 2 D[coszz11 sin D[ sin z 2 sin( zz 21 ) 22 D[sinzzz121 sin( D[ ) cos z 1 cos( zz 21 ) 3 D[sin(zz21 )cos( zz 21 ) G z 5 , 23 zwz422212 Esin D[ cos z D[ sin( zz ) D[ cos zz sin 2 2 D[sinzz12 cos D[ sin z 2 sin( zz 21 ) D[22sinzzz sin( ) D[ cos z cos( zz ) 121 2 21 (6.3.3) 3 D[ sin(zz21 )cos( zz 21 ), zz56 , 2 zz65 Z . It is important to write the system of equations (6.3.3) under the form zAzgznnppn (), np, 1,...,6 , (6.3.4) with 178 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS gznn() Bppn sin z Cpp cos z DzzEzznpqsin( p q ) npq cos p sin q (6.3.5) FzzzGzzznpqrsin p sin( q r ) npqr cos p cos( q r ) Hzzzznpqpqsin( )cos( pq ). In the above expressions (,npqr ,, 1,2,3,4), and the summation convection is used over these indices. The nonzero constants of (6.2.5) are given by AA13 1,24 1, Z2 AAA35 G,1,56 65 2 , Z1 BwBw31 ,,42 E (6.3.6) 22 CC31 D[,,42 D[ 33 DD321 D[,,421 D[ EE312 D[,,311 D[ EE422 D[,,421 D[ 22 FF3221 D[,,3121 D[ 22 FF4221 D[,,4121 D[ 22 GG3121 D[,,4221 D[ 33 HH321 D[,.421 D[ Inserting (6.2.19) into (6.3.5) and neglecting the terms up to the sixth order, we have gz1 () 0, gz2 () 0, 222 gz312112() (D[ wz ) D[D[ z( c 1) z D[ 2 (1 czz ) 22 3322 D[(1)(cz 21 D[ a D[ cwaczc D[) D[ (31) [ zz12 32 2 3 2 2 4 D[3()(2cz12 z D[[ c a z2 D[ d c a) z1 2232222 D[(42)adczz 12 D[ (6 dcazz 6)12 232433 5 D[(5adzz 4 )12 D[ ( dazbwacdz 2 )2 ( D[ D[) 1 23 4 3 23 D[[[(5ac 5 d d ) z12 z D[ 10 ( d ac ) z 21 z 23 233 4 D[(10 [dacaczzdaczz 10 [ )12 5 D[ ( ) 12 (6.3.7) 22 5 D[[(),dacbz [ 2 THE COUPLED PENDULUM 179 222 gz41() D[ z ( D[E wz ) 2 D[ (1 cz ) 1 2 D[ ( c 1) zz 12 222332 D[(1 cz )211 D[ ( [ c az ) 3 c D[ zz2 22 3 32 4 D[czzaccwazadz(3 [ 1)12 D[D[D[E ( )2 D[ (2 ) 1 232222 D[(4dazz 5 )12 D[ (6 adczz 6 ) 12 223224225 D[(5 a 4 d 2 czz )12 D[ ( dcz )2 D[ ( b [ d [ acz)1 34 2223 D[5 (baczz )12 D[ (10 d [ 10 acaczz [ ) 21 323345 D[10 (daczz )12 D[ 5 ( daczz ) 12 E wbz 2, gz56() 0, gz () 0. Using (6.2.7), the system of equations (6.2.4) yields to a simplified system of Bolotin type (Bolotin), we refer to as the problem E2 4 zAzFz ¦ i (), (6.3.8) i 1 where 6 Az a z , ¦ np p p 1 6 F ()zbzz , 1 ¦ npq p q pq,1 6 F ()zczzz, (6.3.9) 2 ¦ npqr p q r pqr,, 1 6 Fz3 () ¦ dnpqr zzzz p q r l , pqrl,,, 1 6 Fz4 () ¦ enpqrlm zzzzz p q r l m , pqrlm,,,, 1 with n 1,2,3,...,6 . The matrix A anp is ª 001000º « 000100» « » «D[ w D[ 00G 0» A « » . (6.3.10) « D[ D[ Ew 00 0 0» « 000001» « 2 » ¬« 0000Z 0¼» The constants of (6.3.9) are defined as follows 180 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS ab 0.16605, 0.00761, cd 0.49670, 0.03705, 3 cacc3111 D[ D[ D[ wa , 232 cc4122 D[(3 [ 1), c 4112 3 D[c , cc3222 D[() [ a 22 22 ddc31111 D[(2 a) ,,dad31112 D[(42 c ) 22 2 ddc31122 D[(6 6a ) , da31222 D[(5 4d ) , 2 2 dd32222 D[(2 a) , da41111 D[(2 d ) , 2 22 dd41112 D[ (4 5a ) , da41122 D[ (6 6d c ) , 22 22 dad41222 D[ (5 4 2c ), dd42222 D[ () c , 33 ebwac311111 D[ D[d , 3 3 ea311122 10D[ (cd ) , ea411222 10D[ (cd ) , 22 3 eac411122 D[ (10 [ 10 [dac ) , ea312222 5(D[cd ) , 22 eacdb322222 D[() [ [ , 22 ea411111 D[[ ()c [db , ew422222 E b . 6.4 LEM solutions of the system E2 In this section we derive the LEM solutions of the system E2 (Toma, Munteanu et al. 2002b). For this, we apply to equations of the system E2 an exponential transform depending on four parameters of the form (1.6.2) vt ,V1 , V 2 , V 3 , V 4 exp( V11z V 22 z V 22 z V 44 z) , Vi R , (6.4.1) that yields a linear first order partial differential equation wwwvv44 4 23 v4 wv ( ¦¦Vnab np V ¦n npq V ¦ n c npqr wt n 11 p wVppq,1 wVwVpqpqr,,1 wVwVwVpqr (6.4.2) 44ww45vv de), V¦¦n npqrl Vn npqrlm pqrl,,, 1 wVpqrl wV wV wV pqrlm,,,, 1 wVpqrlm wV wV wV wV with initial conditions 0000 vz 0,V1 , V 2 , V 3 , V 4 exp( V11 V 22z V 33zz V 44) , (6.4.3a) THE COUPLED PENDULUM 181 and bounding condition at infinity |,,,,|vt VVVV1234 dv 0 as t of, (6.4.3b) where vvtt0 sup{ | ( ,VtV ) | : 0, R }. An important step in solving the equation (6.4.2) is the observation that a series of the form ¦ ()PVt mm verifies the equation. By looking on the explicit form of this m 0 particular solution we can derive, without any information and in a purely deductive way, other series of the form ¦ ()()PPVVttmnmn , which verify the equation. So, there mn,0 must be a family of series that can be taken into consideration. In fact, this observation yields to a solution of (6.4.2) and (6.4.3) under the form 44iij i VVij V vA1 ``kk AA`l ¦¦kk ¦ ¦ l ki 11 ii!!klij,1,1 j! klz 4 ijr ij rVVV AAA ```klm ¦¦kl m (6.4.4) klm,, 1 i , jr , 1 ijr!!! klmzz 4 ijrs ij rsVVVV AAA A ````klmn, ¦¦kl m n klmn,, , 1 i , jrs , , 1 ijrs!!!! klmnzz z where An ()t , n 1, 2, 3, 4 , are defined as At() {( t )kk1 A ( ) ( t , ) ( t ) B ( ) ( t , )}. (6.4.5) nn ¦ PKk)knPK PK\PKkk k ,0K Indices take integers values k 0,1,2... , K 0,1, 2,... . In (6.4.5) the functions )k (,)PKt and \PKk (,)t and P PK() are defined as (,)ttA ()mk1() k (), (,)ttB ()mk1() k (), (6.4.6) )PKkm ¦ P K \PKkm ¦ P K mk 1 mk 1 4 P ¦ Dj Oj , (6.4.7) j 1 4 with ¦Dj K1 , Dtj 0 , j 1, 2, 3, 4 . In (6.4.7) p j , j 1, 2, 3, 4 , are the roots l 1 O112 ppp,,,O 13 O 24 O p2 , (6.4.8) of the characteristic equation (O42p O')0 , (6.4.9) with 182 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS p D[D[E ww , ' D[Ewww D[ 2 E . (6.4.10) The unknowns Ank and Bnk depend on K in the following way ACBBCCnk ()K k () K nk (), K nk () K k () K nk (). K (6.4.11) The constants Ck ()K verify the recurrence relation k 22K C ()K C ()K , (6.4.12) k kk(2 1) k 1 with p |(1i)|*K C ()K , (6.4.13) 0 2! where* is the Gamma function, and *K(1 i ) K2 ||[12 ]1 . (6.4.14) 2 *(2) n 0 (2n ) ()k ()k Constants Am ,Bm are related between themselves by ()k ()k BmAmm()K ' () K, (6.4.15) ()k where Am ()K is defined by K AA()kk 1, () , kk12k 1 (6.4.16) ()kkk () () ()(1)2mkmk Ammm K A12 A , m ! k2. We mention that Bnk ()K and Cnk ()K depend on the initial constants and on the ()k constants abcde,,, , of (3.1.1). Substituting Bm ()K given by (6.4.15) into (6.3.6)2 , we obtain mk1() k \PKk (,)ttB ¦ () Pm () K. (6.4.17) mk 1 We point out that \P(,)t K can be calculated from )P(,)t K by formulae d \P(,)tt K P )P (,) K P)PK (,)t . (6.4.18) dt Denoting k 1 Fkk(,)PKtC ()() K P t ) k (,) PK t, (6.4.19) we obtain for (6.4.5) At() { B ( ) F ( t , ) C ( ) F ( t , )} nn ¦ kKPKKPknkkK . (6.4.20) K,0k THE COUPLED PENDULUM 183 It is easy to observe that F00(,0)sin,PttFtt P (,0)cos. P P (6.4.21) Now, return to (6.4.4), and after a little manipulation, we have VVijVk Vl vA(1ijk12 )(1 A )(1 A3 )(1 Al 4 ) , (6.4.22) ¦¦12 ¦ 3 ¦4 ijijk!!! k ll! and vx ,V1234 , V , V , V exp( V 1122AAAA V V 33 V 44) . (6.4.23) Finally, the solution of (6.3.8) results from (6.4.27) zt() At () { B ( ) F ( t , ) C ( ) F ( t , )} nn ¦ nkk KPKKPK nkk . (6.4.24) k,0K We refer to (6.4.24) as the LEM representations of the solutions of problem E2. These solutions are bounded at t of . We name the functions Fk (,)PKt the incomplete Coulomb functions of vibration, or Coulomb functions of vibrations, since they are similar to Coulomb wave functions (Abramowitz and Stegun). So, the LEM solutions are describable as a linear superposition of Coulomb vibrations (Donescu 2003). The first terms in (6.4.24) represent the linear part of the solutions of the problem E2 (n 1, 2, k 0,1 ) zC1 10cos ptB 1 10 sin ptC 1 11 cos ptB 2 11 sin pt 2 , zC2 20cos ptB 1 20 sin ptC 1 21 cos ptB 2 21 sin p 2t , (6.4.25) where the constants are defined as K 0 ( n 1, 2 , k 0,1,...,9 ), 0 o 0 o zvz221 zvz211 C10 , C11 , D1 D1 0 pp22o 0 pp11o zNz324 zNz314 aa13 13 aa13 13 B10 , B11 , D2 D2 CC20 10v 1 , CCv21 11 2 , BBv20 10 1 , BBv21 11 2 , 22 pp21 pp12() N 2 N 1 D1 , D2 2 , aa32 13 a13 2 2 2 1 p1 1 p2 ap42 1 a41 va13 ()1 , va23 ()1 , Na13 ()1 aa32 13 aa32 13 pa132 a 13 p1 184 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 00pw22 0 p 2 00 pp12 2 zz12() z 1 zz 21 ap42 2 a41 bb412 312 b322 Na23 ()1 , C12 , pa232 a 13 p2 D3 00ppp11o 0012p zz12() wz 11 zz 21 bb312 412 b422 C13 , D3 00pw12o p 1 00 pp12 zz33() z 4 zz 34 bb422 322 b312 B12 , D4 00pp22o 00 p12p zz43() wz 14 zz 34 bb322 422 b412 B13 , D4 CCw22 12 1 , CCw23 13 2 , BBw22 12 1 , BBw23 13 2 , 22 bppbpp312 2 1 422 1 2 pp1 2() b 312 pw 2 1 b 422 pw 1 2 D3 , D4 , bb322 412 bb312 422 2 2 3 p1 3 p2 wb14 ()12 , wb2 ()412 , bb312 322 bb312 322 0 02 pw24 02 p2 0 02 03 03 zz12() z1 szz121 sz 21 sz 32 cc4111 3222 cc4111 3222 C14 , D5 002 pw23 02 p2 020 03 03 zz21() z2 szz421 sz 51 sz 62 cc3111 4222 cc3111 4222 C15 , D5 0 02 pw14 02 p1 0 02 03 03 zz3() 3 z4 (szz134 sz 23 sz 34)D 1 cc3222 4111 cc 3222 4111 B14 , D6 0 02 pw23 02 p2 02 0 03 03 zz43() z4 (szz434 sz 53 sz 64)D 2 cc3222 4111 cc 3222 4111 B15 , D6 CCw24 14 3 , CCw25 15 4 , BBw24 14 3 , BBw25 15 4 , THE COUPLED PENDULUM 185 22 1 cc3112 4122 ppcc 2 1 3122 4112 pp 1 2 D5 . 16 cb3111 412 cb 4222 322 The following theorem holds: THEOREM 6.4.1 The system of equations E2 dz NN N n az b zz c zzz ¦¦np p npq p q ¦npqr p q r dt p 1,1,pq pqr,1 NN dzzzz e zzzzzn N ¦¦npqrl p q r l npqrlm p q r l m , 1,2,3,..., , pqrl,,, 1 pqrlm,,,, 1 with initial conditions 0 0 zznn(0) , zRn , admit for [d0.3 , bound-stated solutions at t of , of the form zt() { B ( ) F ( t , ) C ( ) F ( t , )}, nn ¦ kknKPKKPkkK k,0K where k 1 Fkk(,)PKtC ()() KP t ) k (,) PK t, are Coulomb functions of vibration, and (,)ttA ()mk1() k (), )PKkm ¦ P K mk 1 k 22K p |(1i)|*K CC()K ()K, C ()K , kkkk(2 1) 1 0 2! K AA()kk 1, () , kk12k 1 ()kkk () () ()(1)2mkmk Ammm K A12 A , m ! k2. 6.5 Cnoidal solutions The aim of this section is to apply the cnoidal method to both problems E1 and E2. First we analyze a particular case which can reduce the equation of motion to a Weierstrass equation of the type (1.4.14) that admits an analytical solution represented by a sum of a linear superposition and a nonlinear superposition of cnoidal vibrations. We consider the uncoupled case D D ( I12 II ) °T11wsin TD cos T 1 0, (6.5.1) ® ¯°TE222wsin TD cos T 0, 186 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS where Mgl2 MMgl 11 ww,,122 E Ik Ik a alM M [,,11 D m, lI M2 and initial conditions 0000 T1122112(0) TT , (0) TT , (0) TTp , (0) Tp2 . (6.5.2) Multiplying the first equation by 2T1 , and the second one by 2T2 , and integrating we obtain 2 2 T111 2coswC TD 2sin T 1 , T222 2cos2sinwC E TD T 2 , (6.5.3) whereCi , i 1, 2 , are integration constants. Approximating the trigonometric functions by series of five-order, (6.5.3) yields 2 Tiii P () T , i 1, 2 , (6.5.4) wherePii()T are polynomials of fifth-order in Ti 2345 Paaaaaaiiiiiiiiiiii()T 012345TTTTTi , i 1, 2 , (6.5.5) with awCawCa01 2,1 02 2E2 , 11 D 2, aA 2,D a 2 wc , a 2E wc , 12 21 22 (6.5.6) aaaa31 2, D32 2, D awdawd41 2,42 2E , aba51 2, D52 2, Db where, for sake of simplicity, we take 2,2wC 12 E wC , aa11 12 20 Dz . We recognize in (6.5.4) the Weierstrass equations of the form (1.4.14), for n 5 2 2345 T A12345 TAAAA T T T T, (6.5.7) where, by dropping the second index for constants a 13 5 A aAaA,, aAaA ,2, a. 112233445522 2 The assumed initial conditions for (6.5.7) are T(0) TT0 , (0) Tp0 . (6.5.8) THE COUPLED PENDULUM 187 We know that the equation (6.5.7) admits a particular solution expressed as an elliptic Weierstrass function that is reduced, in this case, to the cnoidal function cn 2 G(;,)(tggeeecc23 2 2 3 )cn( eet 1 3 G) , whereGc is an arbitrary constant, eee123,, , are the real roots of the equation 3 40ygyg2 3 with eee12!!33 , and g2 , g are expressed in terms of the constants 3 2 Ai , i 1,2,...,5 , and satisfy the condition 2 !27gg 3 0 . The general solution of (6.5.7) is expressed as a linear superposition of cnoidal vibrations as given by (1.4.13) n cn2 [tm ; ] , Tlin ¦ D l Z l l l 1 where0 ddml 1 , and the angular frequencies Zk , amplitudes Dk depend on the initial conditions. The Weierstrass equations admit also solutions expressed as a nonlinear superposition of cnoidal vibrations. To derive these solutions, we adopt the Krishnan solution (Krishnan) O()t T()t , (6.5.9) int 1(Pt) where ()t is the Weierstrass elliptic function given by (6.5.8), and O , P , are arbitrary constants. Substituting (6.5.9) into (6.5.7) we obtain four equations in OP , , g2 and g3 24 3223 4 OP2 A12 PAA OP 3 OP A 4 OP A 5 O, (6.5.10a) 3223 44OP A12 P 3AAA OP 2 3 O P 4 O , (6.5.10b) 3 66O OP22g AAA P 3 OP O2 , (6.5.10c) 2 21 2 3 2 OPg231 2 OPgA P A2 O . (6.5.10d) From (6.5.10a), (6.5.10b) we get 432234 65AA12P OP 4 A 3 OP 3 A 4 OP 2 A 5 O 0 . (6.5.11) Let us consider the special case where (6.5.11) is reducible to ()RSO P4 0 , (6.5.12) 1/4 §·A5 P ¨¸ O, (6.5.13) ©¹3A1 with 188 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 4 3 4 RA (2 )1/4 , SA (6 )1/4 , A RS 3 , A RS22, A RS3 . (6.5.14) 5 1 2 5 3 2 4 3 We observe that both quantities RS and in (6.5.12) are both real or imaginary. In the last case this equation leads to i(RSccO P)4 0 with Rc and Sc real quantities. 1 1 We calculate Aa Dz0 for the first equation (6.5.5), and Aa Dz0 , 112 112 A 5 for the second equation (6.5.5), with D!0 . In both cases we have 5 b !0 . 33A1 Then 1/4 §·5b P ¨¸ O. (6.5.15) ©¹3 Using (6.5.13) we have a unique constant O from (6.5.10a) and (6.5.10b) 3/2 O 30(3AA15 ) . (6.5.16) The definition of the constant A5 yields 5 5 A ab D5 , or A ab 5D . 552 552 From (6.5.16) we have for both situations O 30(15 D23/b ) 22, D b ! 0 . (6.5.17) From (6.5.15) and (6.5.16) we obtain 1/4 §·5b 23/ 2 P 30¨¸ (15Db ) . (6.5.18) ©¹3 The unknowns g2 and g3 are computable from (6.5.10c) and (6.5.10d). It results that OP, , g2 and g3 are always real. The expression (6.5.9) becomes O[(eee )cn(2 eet )] T()t 223 13 , (6.5.19) nonlin 2 1[Peee223 ( )cn( eet 13 )] whereOP and are given by (6.5.17) and (6.5.18). A general form for nonlinear part of the solution of (6.5.7) can be written as n cn2 [tm ; ] ¦EZkkk k 0 Tint (,)xt n . (6.5.20) 1cn[;2 tm O¦ kk Z k ] k 0 So, we can conclude that the solution of the equation (6.4.7) and the arbitrary initial conditions (6.4.8) consist of a linear superposition of cnoidal vibrations and a nonlinear interaction between them. THE COUPLED PENDULUM 189 Figure 6.5.1 Solutions z1 and z2 calculated by cnoidal and LEM methods. Next, we show that the cnoidal method is suitable for solving the system of equations E1. Suppose the representations of solutions ztk () , k 1, 2, 3, 4 , of the form w2 zt( ) 2 log4KKK()k ( , ,..., ) , (6.5.21) knwt 2 12 n where the 4 function is defined by f nn ()kk() 1 ()kk () () k 4KKKnn(12 , ,..., ) ¦¦ exp( iMMBjKji ¦ ijM j ) , (6.5.22) ()k ji1,2 j1 Mi f 1ddin with K j Zjjt E , 1 ddjn. (6.5.23) Here, k j are the wave numbers, Z j are the frequencies, E j are the phases, and n the number of degrees of freedom for a particular solution. The following result holds: THEOREM 6.5.1 The bound-state solutions ztk (), k 1, 2, 3, 4 , of the system E1 given by (6.2.20), (6.2.21) can be written as ()kk () ztk() z lin (KK ) zint ( ), K [...KK12 Kn ] , (6.5.24) ()k wherezlin represents a linear superposition of cnoidal vibrations n z()k 2cn[;2 tm ] , (6.5.25) lin ¦Dlk() Z lk () lk () l 0 190 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS ()k andzint represents a nonlinear interaction among the cnoidal vibrations n cn2 [tm ; ] ¦EZlk() lk () lk () ()k l 0 zxtint (,) n , (6.5.26) 2 ] 1cn[;O¦ lk() Z lk ()tm lk () l 0 where0 ddm 1 ,Zk and Dkk, E , Jk are determined from initial conditions (6.2.18). In the small oscillations case M121212 MllIII,, (D D , E 1) the solutions for m 0 are given by zC1101101112112 cn ptBptCptBpt sn cn sn , zC2 20cn ptBptCptBpt 1 20 sn 1 21 cn 2 21 sn 2 . The equivalence between LEM and cnoidal solutions for the problem E2 ([d0.3 ) is shown by numerical treatment of differential equations (Collatz, Halanay, Scalerandi et al.). Figure 6.5.2 The transient solutions z1 and z2 of the pendulum. THE COUPLED PENDULUM 191 Figure 6.5.3 Stabilized solutions z1 and z2 of the pendulum. a Consider an example g 10m/s2 , ll , M 10kg , for which [ 0.25 , 122 2 m 1.5 , k 70 . In Figure 6.5.1 are represented these solutions. As a result the solutions calculated by LEM and cnoidal methods coincide in a proportion of 97.7%. No visible differences are observed. Figure 6.5.2 displays the transient evolution of solutions zt1 () andzt2 () form 1 , 4 [ 0.9 , k 1.6u 10 , and the initial conditions zz12 0.8, z3 z4 0.02 , calculated by the cnoidal method. Figure 6.5.3 shows the stabilized solutions zt1 () and zt2 ()fort ! 100 . 6.6 Modal interaction in periodic structures Davies and Moon have studied in 2001 an experimental structure consisting of nine harmonic oscillators coupled through buckling sensitive elastica. The structure was modeled by a modified Toda lattice, and analytical results confirm the soliton-like nature of waves observed in the structural motions. These systems exhibit complex nonlinear wave propagation including solitons. The modal interactions and the modal trading of energy were studied in the early 1950s by Fermi, Pasta and Ulam. Elastic models of crystal lattices are developed by Born and Huang, Teodosiu. The Toda 192 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS analytic solutions for a lattice with exponential interactions take the form of both periodic traveling and solitons (Toda). Consider the lattice model shown in Figure 6.6.1. This model is used by Davies and Moon for modeling the experimental structure, for N 9 . In the figure the masses are m , all the spring constants are k , and all the damping coefficients are c . Figure 6.6.1 Nonlinear lattice model. The equations of motion of the lattice model shown in Figure 6.6.1 are ck1 xxxxxFxxFxx (3 )[()()] , (6.6.1) jjjjjjjjmm11 m1 j1 withj 1,2,..., N , and boundary conditions of zero displacement of the fixed end and zero force on the free end x0 0 , xNN1 x , (6.6.2) wherem is the block mass, k is the cantilevered beam stiffness, c the damping coefficient, F represents the nonlinear force-displacement relationship for the buckling elements, and N the number of masses. The damping term 3x j in the motion equation comes about because the dampers between masses and the ones attached between the masses and ground are taken to have identical coefficients. The expression for the force- displacement behavior of the buckling elements is the same as that used by Toda to model the interaction forces in an atomic lattice (Toda and Wadati, Toda) J Fx( x ) [exp( bx ( x )) 1] , (6.6.3) jj11b jj whereJ is the element’s linearised stiffness, and b determines the strength of the nonlinearity. For the atoms interaction in a lattice of a solid the Morse interaction force is valid in quantum mechanics of an electron motion, D being a constant FxxF(jj10 ) [exp( D 2 ( xxjj 1 )) 2exp( D ( xxjj1 ))] . Morse force of interaction governs precisely the nearest-neighbor interaction in an anharmonic lattice of the atoms in a diatomic molecule of solids in a continuum limit. But, no exact solutions are found for elasticity problems governed by Morse force. For Toda interaction force there are some exact explicit solutions for the model. With k and c set to zero, equations (6.6.1) and (6.6.3) reduce to a Toda lattice, and the functional form of the nonlinear soliton solution is given by THE COUPLED PENDULUM 193 1 mE2 xx ln[ sech2 (kjtE ) 1] . (6.6.4) jj1 b J Substitution of (6.6.4) into (6.6.1) and (6.6.3) with k and c set to zero, shows that it solves these equations if we have O E sinh k . (6.6.5) m IfJ!0 and b 0 , xtxtjj1 () () 0 for all values of jt and . This corresponds to the compressive atomic lattice waves studied by Toda. If J ! 0 and b ! 0 , then Ea xtx() ()t ! 0 , we have the tensile waves, which propagate to the speed c , j1 j k wherea is the distance between masses. The speed of this soliton is dependent upon the amplitude. For the case of small displacement and zero damping, (6.6.1) and (6.6.3) reduce to the linear conservative system defined by k J xx(2 xxx ) 0, j 1,2,..., N . (6.6.6) jjmm j11 jj Substituting a periodic traveling wave solution of the form x j (tA ) exp[i( kajtZ )], into (6.6.6) we obtain the dispersion relation 4J ka k Z r sin2 ( ) , (6.6.7) mm2 whereZ is the wave frequency, k is the wave number and a is the spacing between adjacent masses. From (6.6.7) we see that the system allows only in the Brillouin zone of frequencies to propagate without attenuation. The equation (6.6.6) has the solution N (2kj 1) S xtjk() ¦ At ()sin , jN 1,2,..., , (6.6.8) k 1 21N where Ak are time-dependent coefficients. The energy of the k- mode of vibration is (Davies and Moon) 21Nm§· 22 k 2(21) kS EAAkk ¨¸k(2sinJ ). (6.6.9) 42©¹ 2 2(21)N If initial conditions are small the nonlinear model (6.6.1) will have approximately linear behavior, and the modal energies given by (6.6.9) will be nearly constant in time. For large initial conditions, the nonlinearities may lead to complex modal interactions (Duncan et al.). The experimental results of Davies and Moon in studying a periodically reinforced structure with strong buckling nonlinearity, have shown that the modal interaction is characterized by highly localized waves. These waves were shown to be similar to Toda solitons. 194 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS Let us analyze the modal interaction by using the Toda interacting equations. We know that the N- soliton solution of a nonlinear equation can be regarded as a system of Nt- interacting solitons, each of which becomes a single soliton as orf . When we haveN single solitons initially apart in space they interact with each other and become apart without exchanging their identities. To study the interaction modal solitons, we consider (6.6.1) and (6.6.3) under the form of the original Toda equation d()Vn d(1 Vn)(1 Vn)2 Vn(), (6.6.10) 1()Vn or dln[12 'Vn ()]2 V 0, (6.6.11) '2Vn() Vn ( 1) Vn ( 1)2 Vn (). (6.6.12) Here, the operator d/ is d w wt and n f,..., 2, 1,0,1,2,..., f. Yoneyama proposes the following form of interacting Toda equations, which represent a natural extension of the Toda equation (6.6.10) d()Vn d(1i Vn)(1 Vn)2 Vn(), i 1,2,..., N , (6.6.13) 1()Vn ii i with the total wave Vn() is N Vn() Vn (). (6.6.14) ¦ i i1 The equation (6.6.13) is the Toda–Yoneyama equation. In the following we present the explicit form of the solution Vni () given by Yoneyama. We see that summing up (6.6.13) and using (6.6.14) we obtain (6.6.10). The solution Vn() has the form Vn() dln2 f . (6.6.15) The function f is given by fI(J12 ,JJ ,...,N ) det[B (n 1)] , (6.6.16) where I andBNN are u matrices 1 Ikl G kl , Bnkl () IIk () n l () n, Iiii()nC exp()J n, Jii()nt E Di n , (6.6.17) 1 zzkl with arbitrary real constants D and Ck , 1 d , lNd , iN 1,2,..., , and zz1 z rexp( D ) , E ii. iii 2 Let us introduce N -independent time variables ti , i 1,2,..., N , and define as THE COUPLED PENDULUM 195 )iii()nC exp() * n, *iii()nt EDi n , (6.6.18) FAI(**12 , ,..., *Ni ) ( * )det[ B (n 1)] , (6.6.19) whereBNN is a u matrix with elements 1 Bnkl () ))k ()n l () n, 1 d k , lNd , (6.6.20) 1 zzkl n A( ) exp[ ()n ()]n , (6.6.21) *ii ¦ D *Ei i 1 with arbitrary functionsDi ()n and E()n . Introduce the operators dˆ Ft (** , ,..., * ) (ww / )F ( ** , ,..., * ) | . (6.6.22) iNiN12 12 t12 t ... ttN The solution of (6.6.13) is given by (6.6.15), where Vn() is Vn()ddlnˆ F ddlnˆ F. (6.6.23) ¦¦¦k k kkk N Here fFˆ and ˆˆ . Substituting (6.6.23) into (6.6.12) we have dd ¦ k ddii d¦ dk k 1 k 1 d0ˆ G , G dln[1( d)ˆ 22F ] d lnF . (6.6.24) ¦ i ¦¦kl ' ¦ m i kl m ˆ This equation is satisfied if G is a constant and if diG 0 for each i . So, we must have dˆˆˆ d ln[1 ( d )22F ] d d ln F , (6.6.25) ik¦¦ l i' ¦ m kl m or d(ddˆ lnF ) d(di '2 dln)ˆ F . (6.6.26) 1()Vn i Therefore, the solution of (6.6.13) is NN Vnˆ F Vn. (6.6.27) () d¦¦ dlni i () ii 11 The explicit form of Vni ()is obtained by defining a symmetric NNu matrix \(,nm ) and three diagonal N u N matrices EO, and Z of elements 1 <(,)nm < (,) nm [()()IIII n m ()()] m n , (6.6.28) ij ji 2 i j i j zz1 E GE, O GO, Z G z , O i i . (6.6.29) ij ij j ij ij j ij ij j i 2 196 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS Also define the N- component column vector I()n of elements Iiii()nC exp()J n. In these conditions, the identities hold dBn ( 1) E Bn ( 1) Bn ( E 1) , d(Bn 1) < (, nn 1). (6.6.30) Also, we have <zII(,nn 1) () nT ( n 1), (6.6.31) T where the transpose of I()n is I()nnn (II12 (),(),..., IN ())n , and tr[(1)(,Pn< nn 1)(1)]tr[(1)() Pn PnII nT ( n 1)(1)] Pn , (6.6.32) I Pn() . (6.6.33) I Bn() THEOREM 6.6.1 (Yoneyama) The explicit form of Vni ()is given by ˆ Vnii() ddln F 2E i [ Pn ( 1)d( Bn 1)( Pn 1)]ii T 2[(1))()(1)(1)]EiiPn I n I n Pn i (6.6.34) TT 2[(1)()(1)(1)]2EPn II n n P n ii E i Pn i (1)()(1)(1) Ii nPn i Ii n . Proof. From (6.6.33) and taking account that for any matrix A , d(dA11 AAA) 1 , (6.6.35) it results I dlndet(1)tr{dln[(1)]}tr[Pn I Bn d(1)Bn ] . (6.6.36) IBn(1) From (6.6.30) and (6.6.33) it results IB(1)n tr[ dBn ( 1)] 2 [ ] 2 [I Pn ( 1)] . (6.6.37) ¦¦ Emmmm E mm IBn(1)mmIBn (1) The equations (6.6.15), (6.6.16), (6.6.33), (6.6.36) and (6.6.30) give Vn( ) d 2 [ I Pn ( 1)] 2P ( n 1) ( n ) ( n 1) P ( n 1) ¦¦Emmmm EIImkkllm. (6.6.38) mm From (6.6.38) we obtain Vn() 2r[ tEII Pn ( 1)() nTT ( n 1)( P n 1)] (6.6.39) 2¦ EiiPn ( I 1) i ( nPn ) i ( I 1) i ( n 1). i Finally, (6.6.40), (6.6. 20) and (6.6.22) yield ˆ d(1)(iklBn EGGi ik ilkl )(1Bn ) , dlndet(iiPn 1)2 E [ I Pn ( 1)]ii . (6.6.40) Chapter 7 DYNAMICS OF THE LEFT VENTRICLE 7.1 Scope of the chapter The heart consists of two pumps (the right and the left) connected in a series that pump blood through the circulatory system. The left ventricle generates the highest pressures in the heart, about 16 kPa, which is four times the pressure developed by the right ventricle (Taber). The left ventricle receives the most attention in the literature because most infarcts occur in this chamber. The left ventricle is a thick-walled body composed of myocardium between a thin outer membrane (epicardium) and an inner membrane (endocardium). The dynamics of the left ventricle is the result of the contractile motion of the muscle cells in the left ventricular wall. Heart muscle is a mixture of muscle and collagen fibers, coronary vessels, coronary blood and the interstitial fluid. The fibers wind around the ventricle, and their orientation, relative to the circumferential direction, changes continuously from about 60D at the endocardium to –60D at the epicardium. This anisotropy influences the transmural distribution of wall stress. Huyghe and his coworkers, Van Campen, Arts and Heethaar, developed a theory of myocardial deformation and intramyocardial coronary flow. In this theory the tissue is considered as a two-phase mixture (a solid phase and a fluid phase representing the different coronary microcirculatory components). A central problem in modeling the dynamics of the heart is in identifying functional forms and parameters of the constitutive equations, which describe the material properties of the resting and active, normal and diseased myocardium. Recent models capture some important properties including: the nonlinear interactions between the responses to different loading patterns; the influence of the laminar myofiber sheet architecture; the effects of transverse stresses developed by the myocytes; and the relationship between collagen fiber architecture and mechanical properties in healing scar tissue after myocardial infarction. We consider in this chapter the cardiac tissue as a mixture of an incompressible solid and an incompressible fluid. Following studies by Van Campen and his coworkers, Huyghe, Bovendeerd and Arts, we construct a model in which the constitutive laws are specified within the broad framework of the intrinsic assumptions of the theory. The cnoidal method is applied to solve the set of nonlinear dynamic equations of the left ventricle. By using the theta-function representation of the solutions and a genetic algorithm, the ventricular motion is describable as a linear superposition of cnoidal pulses and additional terms, which include nonlinear interactions among them. 198 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS In the last section, the evolution equations governing the human cardiovascular system and distortions of waves along this system for perturbed initial conditions, which are responsible to the energy influx conditions, are analyzed. The formation of asymmetric solitons from initial data under and above threshold is the principal topic of this analysis. We refer the readers to the works by Van Campen et al. (1994), Munteanu and Donescu (2002), Munteanu et al. (2002a, c), and Chiroiu V. et al. (2000). 7.2 The mathematical model The underformed heart, in a stress-free reference state, is modeled as a super ellipsoid surface S defined by the implicit equations (Bardinet et al. 1994) c2 22 2 ªºc1 §·xycc22 §· §· zc1 «» 1, (7.2.1) «»¨¸ ¨¸ ¨¸ ©¹aa12 ©¹ ©¹ a 3 ¬¼«» where the constants aii ,1,2, 32 and cii ,1, , are given by cc12 0.773 , aa12 0.892Ra , 3 R, (7.2.2) with R 0.0619m for a particular heart considered in this paper. For a sphere we have cc12 1 and aaaR123 . Representation of the cylindrical coordinates is given in Figure 7.2.1. The z -axis corresponds to the z- axis of inertia of the super ellipsoid model. The muscle fibers in the ventricular wall are assumed to be parallel to the endocardial and epicardial surfaces. The governing equations of the dynamics of the left ventricle are presented by following the above-mentioned papers. Cardiac muscle is considered to be a mixture of two phases, a solid phase and a fluid phase. The equations are derived from the general equations of the continuum theory of mixtures (Truesdell). Nomenclature : V , actual volume of the heart, x (,,)rzT, spatial cylindrical (Eulerian) coordinates, centred in O , §·x §·r ¨¸ ¨¸ ¨¸y oT ¨¸, ¨¸ ¨¸ ©¹zz ©¹ X i , i 1, 2, 3 , material cylindrical coordinates (Lagrangian coordinates), corresponding to a reference state which may be subject to an initial finite deformation, t , time coordinate, DYNAMICS OF THE LEFT VENTRICLE 199 Vs , effective Cauchy stress in the solid representing the stress induced by the deformation in the absence of fluid and measured per unit bulk surface, p , intramyocardial pressure representing the stress in the liquid component of the bi-phases mixture, V Vs pI , total Cauchy stress tensor in the mixture, V VT , u , displacement vector §·uu1 §·r ¨¸uu ¨¸, ¨¸2 o ¨¸T ¨¸ ¨¸ ©¹uu3 ©¹z q , Eulerian spatial fluid flow vector, K 0 , permeability tensor of the underformed tissue, N b , averaged porosity of the underformed tissue, cc , volumetric modulus of the empty solid matrix, H u , displacement gradient, F , deformation gradient tensor F 1H , JF det! 0 , Jacobean of the deformation, C , isotropic energy function defined as cc CJ (1)2 , 2 which is zero in the underformed state and positive elsewhere, W , strain energy function, zero in the underformed state and positive elsewhere, K , permeability tensor given by J 1 K (1)20K , N b E , Green–Lagrange strain tensor defined as 1 EFFI ()T , 2 SEt(,), effective second Piola–Kirchhoff stress tensor SJFF 11Vs ()T , SS T , split into two components SS ap S, SEta (,), active stress tensor, SEtp (,) , the passive stress tensor, split into two components SSp csS , SEc (), component of the passive stress tensor resulting from elastic 200 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS volume change of the myocardial tissue, zero in the underformed state, SEts (,), component of the passive stress tensor resulting from viscoelastic shape change of the myocardial tissue, SEe (), anisotropic (orthotropic) elastic response of the tissue. S e is zero in the underformed state, Gt(), a scalar relaxation function, T a , first Piola–Kirchhoff active stress (not symmetric) related to the second Piola–Kirchhoff active stress by SFTaa 1 , l , current sarcomere length, dl v , velocity of shortening of the sarcomeres v , dt Ihelix , angle between the muscle fiber direction and the local circumferential direction, varying from 60D at the endocardium through 0D in the midwall layers to – 60D at the epicardium, while Itrans is kept zero (Figure 7.2.1), Itrans , angle between the local circumferential direction and the projection of the fiber on the plane perpendicular to the local longitudinal direction, varying from 13.5D at the base through 0DD at the equator to –13.5 at the apex, ww1 w (, ,) gradient operator with respect to the current configuration. wwTwrr z Figure 7.2.1 Cylindrical coordinates and definition of the angle Ihelix , Itrans (Van Campen et al.) The equations of the beating left ventricle are composed from: 1. The equilibrium equation of the deformed myocardium (by neglecting the inertia forces) Vs p 0 . (7.2.3) DYNAMICS OF THE LEFT VENTRICLE 201 2. Darcy’s law in Eulerian form (by neglecting the transmural pressure differences across blood vessel walls) qKp , (7.2.4) J 1 with K (1)20K , the parameters K 0 andN b being specified. N b 3. Continuity equation (conservation of mass) uq 0 . (7.2.5) 4. Passive constitutive laws wC S c , (7.2.6) wE where SS ap S, SSSp c s , (7.2.7) cc CJ (1)2 , JF det! 0 , (7.2.8) 2 SEt(,) JF11Vs () F T , SS T , (7.2.9) 1 EFFI ()T , F 1H , H u . (7.2.10) 2 C is the isotropic energy function, E is the Green–Lagrange strain tensor, and SEt(,) the effective second Piola–Kirchhoff stress tensor, split into an active stress SEa (,t)( and a passive stress SEp ,t) . The passive stress tensor is split into a component resulting from elastic volume change of the myocardial tissue SEc () , and a component resulting from viscoelastic shape SEts (,) described in the form of quasi- linear viscoelasticity (Fung) as t d SGtSs ³ ()WWe d, (7.2.11) f dW wW S e , (7.2.12) wE whereS e is the anisotropic elastic response of the material, Gt()is a scalar function (reduced relaxation function) derived from a continuous relaxation spectrum, W is the potential energy of deformation per unit volume (or elastic potential). The form of the strain energy CC (equation 7.2.4) and W are chosen so that and W are zero in the unstrained state and positive elsewhere, and SEc () andS e are zero in the underformed state. The expression (7.2.4) satisfies those conditions. 202 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS Some expressions for W satisfying the above conditions were proposed in literature: the full orthotropic behavior, the transversely isotropic behavior with respect to the fiber orientation (Bovendeerd et al.), W is an exponential function of Eij (Huyghe, Fung). Because passive ventricular myocardium is nearly incompressible, biaxial tissue testing is a valuable method for characterizing its material properties. Demer and Yin were the first to report the results of such measurements on passive myocardium. They demonstrated that biaxial loaded passive myocardium behaves like a nonlinear, anisotropic, viscoelastic material that can be approximated as pseudoelastic. Huyghe and coworkers concluded that quasi-linear viscoelasticity does not adequately describe the material properties of passive ventricular myocardium. Therefore, resting myocardium is frequently modeled as a finite elastic material using a hyperelastic pseudo-strain function. In this work we have investigated four approaches for the functional form of W : 1. The polynomial function for a subclass of transverse isotropy (Humphrey et al., Huyghe et al. 1991, Bovendeerd et al.) 23 2 Wc 12314151(1)(1)(3)(3)(1)(3) D c D cl cl D cl , (7.2.13) wherel1 is the first principal strain invariant and the transversely isotropic invariant D is the extension ratio in the fiber direction lE1 2tr 3, D 2E jj 1 , (7.2.14) whereEij is the Lagrangian Green's strain. 2. The transversely isotropic 3D strain energy function (Guccione et al., Guccione and McCulloch, Fung, Huyghe) C WQ (exp 1) , (7.2.15) 2 where 222 QbEbE 12ff (2)2(cc E rr EE cr rc bEE3fc cf EE fr rf ) , (7.2.16) with Eij representing strain components referred to a system of local fiber (f), cross- fiber in-plane (c) and radial (r) coordinates. 3. The form developed by Costa et al., Holmes et al., C WQ (exp 1) , (7.2.17) 2 where 222 QcEcEcEcEEcEEcEE 123ff ss nn 222 4fs sf 5fn nf 6sn ns . (7.2.18) The material parameters cc12, and c3 represent the stiffness along the fiber axis, the c sheet axis and the sheet normal axis, respectively. The ratio of 2 governs anisotropy c3 in the plane normal to the local fiber axis, with unity indicating transverse isotropy. The DYNAMICS OF THE LEFT VENTRICLE 203 parameter c4 represents the shear modulus in the sheet plane, and c5 and c6 represent shear stiffness between adjacent sheets. 4. The expression of the ion-core (Born–Mayer) repulsive energy (Delsanto et al., Jankowski and Tsakalakos) 0.5 W ³ DIEI()exp[()R ]dV , (7.2.19) V V where DI((x)) is the repulsive energy function, E(())I x the repulsive range function and V is the heart volume, and 222 Rxxyyzz ()()(000 ) , with (,x00yz, 0 )an arbitrary point. We suppose that DI()and EI()depend on the angles angle Ihelix Itrans in the form DI() D12 ( Ihelix ) D ( I trans ), EI() E12 ( Ihelix ) E ( I trans ). (7.2.20) We mention that the values of the angles depend on the position x . A common problem with estimating parameters of the strain energy function, whether directly from isolated tissue testing or indirectly from strains measured in the intact heart, is that the functions are nonlinear. The kinematic response terms, whether principal invariants (7.2.13) or strain components (7.2.15), (7.2.16), usually co-vary under any real loading condition. This is probably the main factor responsible for the very wide variation in parameter estimates between individual mechanical tests that is typically reported. Although the first three equations are well motivated, a difficulty with any constitutive model based on biaxial tissue tests is uncertainty as to how the biaxial properties of isolated tissue slices are related to the properties of the intact ventricular wall. Though the energy function (7.2.19) is referring to metallic bilayers and noble metals, this form might be more recommendable from a practical point of view. We refer to the fact that the parameter estimation could be greatly improved. By separating the volume change from fiber extension modeled by the repulsive energy function, and shearing distortions modeled by the repulsive range function EI((x)) , the resulting set of response terms should provide an improved foundation for myocardial constitutive modeling. Also, it is an easier identification of only two functions by using a genetic algorithm based on the inversion of the experimental data. In this chapter we adopt the pseudopotential energy approach and consider for W the expression of the ion-core (Born–Mayer) repulsive energy. 5. Active constitutive laws (Van Campen et al., Arts et al.) TTAtlvaa 0 (,, ), (7.2.21) whereT a is the first order Piola–Kirchhoff non-symmetric active stress tensor, related to the second Piola–Kirchhoff active stress by SFTaa 1 , and T a0 is a constant associated with the load of maximum isometric stress. 204 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS The stress tensor T a is convenient for some purposes; it is measured relative to the initial underformed configuration and can be determined experimentally. The cardiac muscle is striated across the fiber direction. The sarcomere length l (the distance between the striations) is used as a measure of fiber length. The experiments show that the active stress generated by cardiac muscle depends on time t , sarcomere length l and velocity of shortening of the sarcomeres dl v . The active stress generated by the sarcomeres is directed parallel to the fiber dt orientation. The function A(,,tlv ) represent the dependency on tl, andv . Figure 7.2.2 Representations of surfaces 61 - endocardium, 62 - epicardium and 63 62 - the portion where only radial displacement is allowed (adapted from Van Campen et al.) We suppose that A(,,tlv )has the form A(,,tlv ) f () tglhv () ( ). (7.2.22) Functions A(,,tlv ),DI() andEI() are determined from experimental data (Bardinet et al. 1994, 1995) by using an optimization approach (Popescu and Chiroiu). A genetic algorithm is considered in section 7.4. The equations (7.2.3)–(7.2.22) represent a coupled set of four nonlinear equations for the displacements uxtk (,) , k 1, 2, 3 , and the intramyocardial pressure pxt(,) that can be written in the form uKp () 0 , ([JFS1sac S S]) F T p 0 , (7.2.23) with J 1 K (1)20K , N b SFTAtlvaas 10(, ,s ), DYNAMICS OF THE LEFT VENTRICLE 205 wC S c , wE t d wW SGts ³ ()WW d, (7.2.24) f dWwE cc 1 CJ (1)2 , EFF ( T I) , 2 2 0.5 W ³ DIEI()exp[()RV ]d . V V The boundary conditions are px(,0) p0 () x , x 61 , k 1, 2, 3 , tT[0, ], 0 uxkk(,0) u , x 61 , k 1, 2, 3 , t [0,T ], (7.2.25) uxk (,0)0 , x 631 6 , k 2,3 , t [0,T ], where 61 is the epicardial surface, and [0,]T the time interval during a cardiac cycle. The cardiac cycle is composed from a systole (contraction of the ventricle) and a diastole (relaxation of the ventricle) phases. We have supposed that at the endocardial surface 61 a uniform intraventricular pressurep0 is applied as an external load (Figure 7.2.2). The loads exerted by the papillary muscles and by the pericardium are neglected. The surface 63 represents the upper end of the annulus fibrosis and is a non- contracting surface with a circumferential fiber orientation. At 6631 only radial displacement u1 is allowed. wW To compute we use the formula wE www1 (XXij) , (7.2.26) wwwExij2 jx i where X i ( X123{{{XX,, YX Z) are the Lagrange coordinates corresponding to a reference state which may be subject to an initial finite deformation, and xi , the final Eulerian coordinates ( x12{{xx,, yxz3 { ) differing from X i by an infinitesimal deformation. It is important to note that, in applying (7.2.26) the Lagrangian coordinates must be considered as constants since they refer to a predefined reference state. For specified form for A(,,tlv ), DI()and EI() the analytical solutions of (7.2.23) – (7.2.26) are determined in the next section. 206 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 7.3 Cnoidal solutions Let us consider the solutions zxtik(,) { u (,), xtk 1,2,3,(,)} pxt , i 1, 2,3, 4, of the system of nonlinear equations (7.2.23)–(7.2.26) under the form zxt(,) zcn (,) xt zint (,) xt . (7.3.1) The first term zcn represents a linear superposition of cnoidal waves and the second term zint , a nonlinear superposition of the cnoidal waves n 2 zxtcn (,) 2¦Dm cn(kxCt mj j m ), j 1, 2, 3 , (7.3.2) m 1 n cn2 (kx Ct ) ¦ Jmmjjm m 1 zxtint (,) n , j 1, 2, 3 . (7.3.3) 1cn(2 kx Ct O¦ mmjjm ) m 1 We employ the usual summation convention over the repeated indices. We take the boundary conditions (7.2.25) 1 under the form 2 pxt000(,) a cn( kxjj Ct0 ), (7.3.4) with kC00j , ,a0 ,j 1, 2, 3 , specified. Consider that n 2 . Experimental calculations have shown us that solutions do not earn any improvements for n ! 2 . For specified form for A(,,tlv ), DI()and EI() the analytical solutions zxi (,t) , i 1, 2,3,4, are given by (7.3.2) and (7.3.3) 2 cn2 (kx Ct ) 2 ¦ Jmi mj j m zxt(,) 2 cn(2 kx Ct ) m 1 , (7.3.5) im ¦Dimjj m 2 m 1 2 1cn(O¦ mikx mj j Ct m ) m 1 whereDmi , kmj , Jmi , Omi andCm , im 1,2,3, 4, 1,2, j 1, 2, 3, are unknown. By introducing (7.3.2) into (7.2.23)–(7.2.25) the unknowns Dmi , kmj , Jmi , Omi and Cm , are easily determined by an identification procedure in terms of the controlling functions M {DI1212 (helix ), D ( I trans ), EI ( helix ), E ( I trans ),ft ( ), glhv ( ), ( )} , (7.3.6) that determine the constitutive laws, the initial data and the constants that appear in governing equations. The analytic expressions (7.3.2)–(7.3.3) of the solutions are available once the controlling functions Mi , i 1,2,...,7 , are specified. The problem to be addressed here DYNAMICS OF THE LEFT VENTRICLE 207 is the inverse of the forward problem. The aim is to use the difference between experimental and predicted by theory parameters to provide a procedure, which iteratively corrects the controlling parameters towards values leading to the least discord between predictions and experimental observations. In the following we consider that the functions Mi , i 1,2,...,7 , are approximated by polynomials of five degree Mii(bb65 ,.... 6i ) ,i 1,2,...7 , characterized by coefficients bj , j 1,2,...,42 . To extract the functions Mii(bb65 ,.... 6 i ) , i 1,2,...7 , from the experimental data, an objective function must be chosen that measures the agreement between theoretical and experimental data 42 4 M -1 1 exp 2 ()P 42¦¦¦4[z()imbzb j im ( j )] , (7.3.7) j 1i1m1 wherezbim() j are the predicted values of the solutions zbi ( j ) , i 1, 2,3, 4, j 1,2,...,42 , given by the forward problem, calculated at M points belonging to the volume between exp the inner and outer wall of the left ventricle. The functions zim are the experimental values of solutions zi measured at the same points. We have extracted these values from the analysis of the volumetric deformation of the left ventricle of the heart, developed by Bardinet et al. The model gives a compact representation of a set of points in a 3D image. Experimental results are shown in time sequences of two kinds of medical images, Nuclear Medicine and X-Ray Computed Tomography. Controlling functions Mi , i 1,2,...,7 , are determined by using a genetic algorithm (GA). GA assures an iteration scheme that guarantees a closer correspondence of predicted and experimental values of controlling parameters at each iteration. We use a binary vector with 42 genes representing the real values of the parameters bj , j 1,2,...,42 (Chiroiu et al. 2000). The length of the vector depends on the required precision, which in this case is six places after the decimal point. The domain of parameters bj [,]aaj j , with length 2a j is divided into at least 15000 equal size ranges. That means that each parameter bj , j 1,2,...,42 , is represented by a gene (string) of 22 bits ( 221 3000000 d 222 ). One individual consists of a row of 42 genes, that is, a binary vector with 22 u 42 components. (1) (1) (1) (2) (2) (2) (42) (42) (42) ()bb21 20 ...bbb0 21 20... b 0 ... b 21 b 20 ...b0 . The mapping from this binary string into 42 real numbers from the range [,]aaj j is completed in two steps: ()j ()j ()j – convert each string ( bb21 20 ...b0 ) from the base 2 to base 10 ()jj () ()j (bb21 20 ...bb02 ) cj , j 1,2,...,42 , – find a corresponding real number bj , j 1,2,...,42 . GA is linked to the problem that is to be solved through the fitness function, which measures how well an individual satisfies the real data. From one generation to the next 208 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS GA usually decreases the objective function of the best model and the average fitness of the population. The starting population (with K individuals) is usually randomly generated. Then, new descendant populations are iteratively created, with the goal of an overall objective function decrease from generation to generation. Each new generation is created from the current one by the main operations: selection, crossover and reproduction, mutation and fluctuation. By selection two individuals of the current population are randomly selected (parent 1 and parent 2) with a probability that is proportional to their fitness. This ensures that individuals with good fitness have a better chance to advance to the next generation. In the crossover and reproduction operation some crossover sites are chosen randomly and exchanging some genes between parents reproduces two individuals. In the newly produced individuals, a randomly selected gene is changed with a random generated integer number by the mutation operation. In the fluctuation operation we exchange a discretized value of an unknown parameter in a random direction, by extending the search in the neighborhood of a current solution. The fitness function is evaluated for each individual that corresponds to the gene representation. The alternation of generations stops when the convergence is detected. Otherwise, the process stops when a maximum number of generations are reached. The procedure consists of the following steps: – The initial population is generated by random selection. – The crossover operator reproduces two new individuals. – New individuals are obtained by the mutation operator. The fluctuation operator extends the search in the neighborhood of a current solution. Figure 7.3.1 Active material behavior as obtained by the genetic algorithm (time dependence of active stress for sarcomere lengths of 1.7, 1.9, 2.1 and 2.3 Pm , length dependence of active stress, and velocity dependence of active stress). The fitness value is evaluated for each individual and in the total population only individuals with a higher fitness remain at the next generation. The alternation of generations is stopped when convergence is detected. If there is no convergence the iteration process continues until the specified maximum number of generations is reached. Next, we report the results of the genetic algorithm. Figure 7.3.1 shows the active material behavior as obtained by GA after 312 iterations. There are shown the time dependence of active stress for sarcomere lengths of 1.7, 1.9, 2.1 and 2.3 Pm , the DYNAMICS OF THE LEFT VENTRICLE 209 length dependence of active stress, and the velocity dependence of active stress. The diagrams are very similar to the active material behavior assumed by Van Campen et al. We consider that the curves are correctly predicted by the genetic algorithm, the results being qualitatively and quantitatively consistent with experimental data given by Bovendeerd et al., Arts et al. Figure 7.3.2 shows the dependence of DI() and EI() on angles Ihelix ,Itrans given by GA after 247 iterations. The distribution of Ihelix and Itrans from the endocardium to the epicardium is also shown. Figure 7.3.2 Angle dependence of DI() D12 ( Ihelix ) D ( I trans ) and EI() E12 ( Ihelix ) E ( I trans ) as given by the genetic algorithm. 7.4 Numerical results The solutions of the equations (7.2.23)–(7.2.26) are represented analytically by using the cnoidal method. The unknown parameters from these representations are determined from a genetic algorithm. The analytical solutions uxi (,)t ,i 1, 2, 3 , and pxt(,) are given by (7.3.5) being describable as a linear superposition of two cnoidal pulses and additional terms, which include nonlinear interactions among them. 210 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS Figure 7.4.1 Initial intraventricular pressure p0 applied at the endocardial surface 61 . Figure 7.4.2 Visualization of displacement field by different values according to the range 0–10 mm. Figure 7.4.3 Location of six representative points belonging to region A Figure 7.4.1 shows the initial pressure p0 given by (7.3.4) applied at the endocardial surface61 (a0 0.34 ). In this way the initial pressure is very similar to the diagram of the pressure in the left ventricle of the human heart by Caro et al. The visualization of DYNAMICS OF THE LEFT VENTRICLE 211 the displacement field by different values according to the range 0–10 mm is shown in Figure 7.4.2. We can see clearly areas on the ventricle where the displacements are high (for example the area A). In Figure 7.4.3 the location of some representative points belonging to the region A are displayed. Figures 7.4.4–7.4.6 represent the time variation of displacements uxtk (,) ,k 1, 2, 3 , during a cardiac cycle calculated in six points, displayed in Figure 7.4.3. Figure 7.4.7 shows the variation of the intramyocardial pressure during a cardiac cycle in the same points. Figure 7.4.4 Variation of displacement u1 during a cardiac cycle in the points displayed in Figure 7.4.3 (points 1,2,3 in the first column and 4,5,6 in the second column) Comparison of these analytical results to the results obtained by Van Campen et al. through the finite element method, shows a good consistency. The analytical solutions allow the possibility of investigating in detail the field of displacements and the field of pressure in each point of the left ventricle. This helps to predict the important features of the left ventricle motion. In the hyperactive region A there are points that move out of phase with the other points (for example point 5). The physician, to help localize pathologies, such as infarcted regions, could use the visualization of these fields. In conclusion, the equations that govern the motion of the left ventricle have the remarkable property whereby the solutions can be represented by a sum of a linear and a nonlinear superposition of cnoidal vibrations. So, we can say that the real virtue of the cnoidal method is to give the elegant and compact expressions for the solutions in the spirit of this property. 212 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS Figure 7.4.5 Variation of displacement u2 during a cardiac cycle in points displayed in Figure 7.4.3 (points 1,2,3 in the first column and 4,5,6 in the second column) Figure 7.4.6 Variation of displacement u3 during a cardiac cycle in points displayed in Figure 7.4.3 (points 1,2,3 in the first column and 4,5,6 in the second column). DYNAMICS OF THE LEFT VENTRICLE 213 Figure 7.4.7 Variation of intramyocardial pressure p during a cardiac cycle in points displayed in Figure 7.4.3 (points 1,2,3 in the first column and 4,5,6 in the second column). 7.5 A nonlinear system with essential energy influx In nature, energy is usually unbalanced and there exists either outflux or influx of energy. The energy influx is not so well understood or modeled (Engelbrecht). The weak energy influx systems that may be called perturbed systems are based on the dilaton concept, which means negative density fluctuations with loosened bonds between the structural elements. The dilatons are able either to absorb energy from the surrounding medium or to give it away (Zhurkov). The mechanism of energy influx yields to the amplification and/or attenuation phenomena due to the energy pumping from one subprocess to wave motion. In mathematical terms the energy influx gives rise to source-like terms in governing equations. For example, a burning candle is a classic case where the velocity of the flame is a nonlinear wave, which depends on the rate of heat release (Engelbrecht). 214 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS The electromagnetic nerve pulse transmission, modeled by Engelbrecht, takes its energy from ion currents. In this section we show that the cardiovascular system can exhibit, in certain conditions, an essential energy influx. A simplified numerical model for the hemodynamic behavior of the cardiovascular system has been developed by P¯evorovsk et al. These authors have studied the pressure generation by chemical reaction in the human cardiovascular system. In this work we improve this model by using a more general description of a transmission line of the pressure pulsations generated and controlled by the chemical reactions. We show that the influx of energy created by small anomalies in energetical equilibrium of the cardiovascular system may change dramatically the picture of the hemodynamic wave behavior. The behavior of the cardiovascular system is characterized by the heart qualities generating the pressure pulsations. The heart performs as four pressure–volume pumps in series propelling the blood flow through the circulatory network. This ability is given by contractivity of the muscular cells creating the heart tissue, which is activated by the chemical energy released as a consequence of the blood and heart muscle metabolism. The released energy during the heart contraction is converted into the mechanical and heat energy. The energy-rich phosphate compounds stored in the heart muscle filaments dominate the cardiac metabolism. Figure 7.5.1 The cardiovascular system model. The key source of the chemical energy is ATP (adenosine triphosphate) reacting both aerobic and anaerobic pathways. The cardiovascular system is modeled by eight elastic segments connected by elastic tubes in a serial circuit depicting both pulmonary (low-pressure) and systemic (high- pressure) blood circulation (Figure 7.5.1). Deoxygenated blood returns from the body through the vens cavae and fills the right atrium (1), which contracts, sending the blood into the right ventricle (2). The right ventricle then contracts, forcing the blood into the pulmonary artery, which carries it to the lungs to pick up oxygen. DYNAMICS OF THE LEFT VENTRICLE 215 Next, the oxygenated blood flows through the pulmonary vein into the left atrium (5), which delivers it to the left ventricle (6). Finally, the left ventricle contracts, ejecting the blood into the aorta and out to the systemic circulation. During the cardiac cycle, each chamber fills during a period of relaxation, or diastole, and ejects the blood during a period of active contraction, or systole. The heart segments passing through the passive and active states during the cardiac cycle have been considered as anisotropic and viscoelastic incompressible material. They act as chemomechanical pumps converting the chemical energy released by the anaerobic hydrolysis of ATP into the mechanical energy and heat. The behavior of the cardiovascular system has been described by the mechanical variables (pressure, volume, flow) characterized by the cardiovascular parameters (compliance, resistance and inductance) and by the physicochemical variables (ATP consumption, chemical work, molar enthalpy). The governing equations of the cardiovascular system are: 1. The pressure–volume relation pFpViiiiiiii D11( J E GVi )1,2,5,6, i , (7.5.1) pViii D22 E i , i 3, 4, 7,8 , (7.5.2) wherept() is the pressure, Vt()the volume, the dot means the differentiation with respect to time. The function F()t and the constants DEii, , Jii, G are defined as ki 2hEi ri Ftii() wt () 1, D1i , Ji , E riiW 2hEi Wi 1 1 Vri E1i , Gi , D2i , E2i , (7.5.3) V0i V0i ci ci whereEh is the Young’s elastic modulus, thickness of the myocardial wall, k parameter of the chemical energy release, r radius of the atrium, V0 initial atrial or ventricle volume, w chemical reaction rate, Vr the residual volume, c the compliance andW relaxation time characterizing the muscle plasticity. The index i denotes the corresponding heart segments. The function wt() has an exponential form and describes all four phases of the cardiac cycle wt( ) w032 (1WWW exp( tc / ))exp( tcc / )exp( t ccc /1 ) , with w0 an initial given constant. The four phases of the cardiac cycle are given by: A. stretching systole-isometric contraction tt (,12t ), tttc ,0, cc t ccc 0 , B. emptying systole-auxotonic contraction ttt (,)23, ttcccccc 2 ,,ttt 0 , C. relaxing diastole-isometric contraction 216 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS ttt (,34 ), ttttttcccccc 23,, , D. filling diastole-noncontractive state ttt (,)41, ttc 0, cc 0,t ccc 0 . We have noted by WWW12,,34 and W the relaxation time of the C phase, of the B phase, of the A phase, and respectively the relaxation time of the D phase. 2. The influence of the pressure pulsations 1 2 GpprGHGij ( i j ij ij i ij ) , (7.5.4) li whereGtij () is the blood flux entering from the i-th to j-th segment, respectively, the blood flux getting out j-th segment to k-th segment, rij the hydrodynamic resistance, li U2 the blood inertia, Hti () 2 with 0.001 the loss coefficient, 2()Ai t 3 3 U 1.062u 10 kg/m the blood density and Ai (t) the flow area. The flow index rule is (ij , ) {(1,2),(2,3),(3, 4),(4,5),(5,6),(6,7),(7,8),(8,1)} . The flow area is given by aGij Ati () , (7.5.5) 2(g ppij ) 60 with a , c an empirical constant, f the heart frequency (beats/min), T the time cTf duration of the cardiac cycle, and g 9.81m /s2 the gravitational acceleration. 3. The continuity equation VGiij G jk . (7.5.6) We consider a set of small-perturbed initial conditions (pressure–time, pressure– volume and chemical reaction rate–time) during one cardiac cycle in the left ventricle, represented in Figures 7.5.2–7.5.4. The values of the used parameters are given in Table 7.5.1 and Table 7.5.2. 5 3 We have considered that Vri 510mu for all i. This set of perturbed data has been obtained numerically starting from the clinical published stable data (Taber, P¯evorovsk et al., Chiroiu V. et al. 2000). We analyze only the evolution of the pressure wave profile with respect to time, due to the perturbed initial conditions in the left ventricle (Figures 7.5.2–7.5.4), during one cardiac cycle. The perturbed data are accepted from the physical and clinical point of view. We present the results of both analytical and numerical integration in order to get information about the pressure wave profiles. The standard Runge–Kutta method of the fourth order is used in the numerical calculation. The transient pressure wave has the form of an asymmetric soliton represented in Figure 7.5.5 (the curve a represents an asymmetric soliton for unperturbed initial data). DYNAMICS OF THE LEFT VENTRICLE 217 We observe that, for perturbed initial conditions, this wave is amplified from its initial form to the certain profile, the amplitude of which reaches the asymptotic value. An intriguing question is the amplification, which is characterized by a fast change in amplitude for small-perturbed initial data. We have analyzed the transient waves taking for the related perturbed initial conditions (Figure 7.5.5, curve b – for Figure 7.5.2, curve c – for Figure 7.5.3 and curve d – for Figure 7.5.4). Figure 7.5.2 The perturbed profile of the pressure–time diagram during one cardiac cycle. The result is that the maximum amplification happens for initial excitations given by the chemical reaction rate during the one cardiac cycle. The changes are both in the amplitude and in the width of the pulse. The broken line denotes the asymptotic value of the pressure amplitude after a large interval of time. The aperiodic time-dependent phenomenon which appears here is in correspondence with the causality principle which states that the transport or propagation processes in the cardiovascular system are due to (causal) chains of interactions (cause–effect) between a source of perturbation (emission of signal) and the response (reception) and this can be realized (due to the inertia of the interacting human body) only after a certain time delay (relaxation), so that the transport occurs at a finite velocity (Kranys). Figure 7.5.3 The perturbed profile of the pressure–volume diagram during one cardiac cycle. 218 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS Figure 7.5.4 The perturbed profile of the chemical reaction rate–time during one cardiac cycle. Figure 7.5.5 The transient profiles of pressure for the perturbed initial data (Figures 7.5.2–7.5.4) in the left ventricle. As a principal conclusion attention has to be paid to the energetic aspects of the cardiac activity. The influx of energy created by small anomalies in energetical equilibrium of the cardiovascular system changes dramatically the picture of the hemodynamic waves behavior. The perturbation of the initial chemical reaction data acts as a stimulus. If the stimulus is below a certain threshold value expressed as ratio between the chemical work and the mechanical work in the left ventricle ( P WWchem/ mec d 2.11557 ), the pressure motion is normal and the stable state returns quickly without any instability in the pulse propagation. If the stimulus is above this threshold, the process turns out to be much more dramatic. In our case we have P 2.3451 . The amplitude of the wave is quickly amplified and the wave exhibits a clear tendency to chaos. A Poincar- map is used to gain further insight into the structure of the chaotic motion of the pressure wave (the case d). Here a Poincar- map is a set of points in the phase plane pt(), pt () plotted at discrete intervals of time, one each cardiac cycle, after 1000 cardiac cycles (Figure 7.5.6). The map reveals the properties of a strange attractor, namely stretching and folding of the sheet of pressure trajectories (Guckenheimer and Holmes). DYNAMICS OF THE LEFT VENTRICLE 219 Table 7.5.1 The quantities used in the governing equations (P¯evorovsk et al.). k [Js/kmol] 3 W [s] 3 V0 [m ] w0 [kmol/m s] W1 [s] W2 [s] W3 [s] 1-A 14 310u -5 0.06 6.44 u 10-5 0.06 0.1 0.006 1-B 400 610u -5 0.01 6.44 u 10-5 0.01 0.03 0.01 1-C 310u 5 710u -5 0.01 5.085 u 10-5 0.06 0.01 0.06 1-D 110u -9 710u -5 0.1 0 0.06 0.3 0.06 2-A 5.4 u 105 1.2u 10-4 0.4 1.65 u 10--4 0.06 0.3 0.006 2-B 5.57 u 104 110u -5 0.14 2.75 u 10--4 0.06 0.3 0.6 2-C 1.6u 103 810u -5 0.03 2.89 u 10--4 0.06 0.01 0.06 2-D 110u -9 1.8u 10-4 0.3 0 0.06 0.1 0.06 5-A 1.4 u 104 210u -5 0.1 7.02 u 10-5 0.06 0.003 0.006 5-B 1 u 104 310u -5 0.1 1.026 u 10-4 0.06 0.1 0.06 5-C 50 710u -5 0.1 8.1 u 10-5 0.06 0.03 1.6 5-D 110u -9 710u -5 0.01 0 0.01 0.03 0.01 6-A 4.4 u 105 510u -5 1.25 8.855 u 10-4 0.06 0.29 0.006 6-B 2.55 u 106 310u -4 0.2 2.818 u 10-3 0.06 0.1 0.6 6-C 1 u 105 1.2u 10-4 0.063 1.449 u 10-3 0.06 0.02 1.1 6-D 110u -9 1.2u 10-4 0.1 0 0.01 0.3 0.01 Table 7.5.2 The constants characterizing the mechanical properties of the CVS (P¯evorovsk et al.). i j 3 2 3 3 3 rij [Pas/m ] l j [Pas /m ] c j [m /Pa] E j [Pa] pi [Pa] Gij [m /s] 12 9.5 u 104 110u 2 - 2.2 u 10-3 7.53 u 104 210u -7 23 1.2 u 105 110u 2 - 3.75 u 10-5 1.33 u 104 110u -7 34 4 u 106 5.5u 103 5.2 u 10-8 --210u -7 45 5.9 u 105 110u 4 5.9 u 10-8 --110u -7 56 1,5 u 105 1.1u 102 - 1.5 u 10-3 - 310u -7 67 5 u 105 310u 2 - 1.2 u 10-5 - 110u -7 784.5 u 107 510u 5 6.8 u 10-9 - 9.33 u 104 210u -7 811.3 u 107 910u 4 6.9 u 10-9 - 3.99 u 104 110u -7 Figure 7.5.6 Poincar- map p(),tpt ()after 1000 cardiac cycles. Chapter 8 THE FLOW OF BLOOD IN ARTERIES 8.1 Scope of the chapter Arteries conduct blood from the heart to the tissues and peripheral organs. The left ventricle pumps blood into the aorta, and the right ventricle pumps blood into the pulmonary artery. These main conduits branch into smaller vessels, and narrow arterioles vary their dimensions to regulate blood flow. Pulsatile flow of blood in large arteries has attracted much attention in blood dynamics. Computational fluid dynamics has emerged as a powerful alternative tool to study the hemodynamics at arterial tubes. Compared with experimental methods, fluid dynamics can easily accommodate changes in blood flow theory. Experimental studies of blood pulses revealed that they propagate with a solitonic characteristic pattern as they propagate away from the heart (McDonald, 1974). Euler in 1775 obtained the one-dimensional nonlinear equations of blood motion through arteries, for the first time. Rudinger (1966), Skalak (1966), Ariman et al. (1974), Yomosa (1987), Moodie and Swaters (1989) have developed further this nonlinear theory. In 1958 Lambert used the method of characteristics to analyze the motion equations of blood. The finite difference method and the finite elements method are used also for the computations of nonlinear blood flow. In this chapter the soliton theory is employed to describe the dynamical features of the pulsatile blood flow in large arteries. The theory is performed for an infinitely long, straight, circular, homogeneous thin-walled elastic tube filled with an ideal fluid, in the spirit of the Yomosa theory (1987). The theory of microcontinuum model of blood developed by Eringen in 1966 and Ariman and his coworkers, Turk and Sylvester, in 1974, is applied next to describe the transient flow of blood in large arteries subject to an arbitrary two-soliton blood pressure. The blood is assumed to be an incompressible micropolar fluid. The flow velocity, the micro-gyration and the cross-sectional area are calculated as functions of the two-soliton blood pressure pulse. The effects of increasing hematocrit on the amplitudes of the flow velocity and of the microgyration are analyzed. The main bibliographies of this chapter are the works of Ariman et al. (1974), Yomosa (1987), Eringen (1966, 1970), Munteanu et al. (1998). INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 221 8.2 A nonlinear model of blood flow in arteries McDonald in 1974 analyzed the form of the flow and pressure waves at certain locations from the ascending aorta to the saphenous artery in the dog (Figure 8.2.1). The propagation of the pressure pulse is accompanied by an increase in amplitude and a decrease in pulse-width which have been noted as peaking and steepening. The increase in amplitude is in accordance with an increase of the pulse–wave velocity and is combined with the generation of a dicrotic wave (Hashizume). Sections 8.2 and 8.3 present the Yomosa theory, performed for an infinitely long, straight, circular, homogeneous thin-walled elastic tube embedded in the tissue. The blood is assumed an incompressible and nonviscous fluid. Figure 8.2.1 The behavior of the flow velocity and pressure pulses from the ascending aorta to the femoral artery (McDonald). Consider the one-dimensional fields of longitudinal flow velocity vztz (,), the blood pressurepzt(,) , and radial displacement of the arterial wall uzr (,)t , expressed in cylindrical coordinates (rz,,)T , where rz is the radial coordinate and , axial coordinate. Assume rotational symmetry and the hypothesis of uniform distributions of flow velocity vzz (,)t and the fluid pressurepzt(,) over the cross-section of the vessel. – Navier–Stokes equation of motion for longitudinal flow wwvv1 wp zzv 0 , (8.2.1) wwUwtzzz whereU is the density of blood. – Continuity equation which expresses the incompressibility of blood wA w()vA z 0 , (8.2.2) wwtz 222 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS where A(,)zt is the cross-sectional area of the tube. The third equation describes the radial motion of the arterial wall. This equation was derived by Yomosa. We present his derivation of the equation. Consider the small segment of the tube wall surrounded by inner and outer surfaces of the wall and parallel two cross-sectional surfaces perpendicular to z -axis defined by constant coordinates of z andz dz , and two surfaces defined by constant angles of T and TTd (Figure 8.2.2). Figure 8.2.2 A small segment of the tube wall and forces acting on it (Yomosa). The radial displacement of the wall ur is defined as (Teodosiu, Solomon) rru 0 r , (8.2.3) where rzt(,) is the radius of the tube, r0 is the equilibrium radius of the tube when the wall is static and pp e , where pe is the pressure outside the tube which can be equal to the atmospheric pressure. The constitutive law of the elastic arterial wall can be written as Vtr Ea H(1 Hr ) , (8.2.4) rr 0 where Hr is the radial strain, E the Young’s elasticity modulus, and a , a r0 nonlinear coefficient of elasticity. The equation of radial motion of the segment considered in Figure 8.2.2 is w2r UT0 (drhz d)2 ( ppet )cos(dd)2 MTVT r l sin(d/2)dhz wt (8.2.5) w VM(sin)dd),zhr T wz l whereU0 is the density of the wall material, h the thickness of the wall in radial direction, Vl the longitudinal extending stress in the direction of the meridian line on the wall surface, Vt the extending stress in the tangential direction, and M(,)zt the INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 223 angle between a tangent to a meridian line on the wall surface and the z -axis, and dl the elementary length of the meridian line in the segment, defined as dclzosdM . For a small angle M , we have the approximations cosM [1ww (rz / )21/2 ] # 1 , sinM # tan M wrz/ w . (8.2.6) These approximations are related to the fact that the term (/)wwzr 2 contributes higher order small quantity in the later perturbation expansions, proportional to H3 , whereas the present theory is carried out by taking into account up to the terms proportional to H2 , where H denotes a small parameter 0 H 1. Thus, the term wV(sin)/l M wz can be approximated as wV( sin M ) / wzE ( w / w z )[((d l d z ) / d zr )( w / wz)] l 1 (8.2.7) 22 2 (3Erzrz1 / 2)(wwww / ) ( / ), whereE1 is Young’s longitudinal elasticity modulus of the wall. This term is proportional toH5 , and therefore can be neglected. Taking sin(d/2)d/2T#T, for small angles dT , we obtain the radial motion equation w2r hV Uhpp ()t . (8.2.8) 0 wt 2 e r We must mention that the real thickness of the wall h in the direction perpendicular to the wall surface is given by hh cos M# h. Also, the inertial effects on the radial motion of the wall caused by the tissue must be considered. So, the effective inertial thickness is H hhc , where h is the thickness of the wall in which the material participates in the elastic deformation and hc is the additional effective inertial thickness of the tissue in which the material does not participate in the elastic deformation. With these assumptions and by considering that the densities of the materials of the wall and the tissue are the same, the equation of radial motion becomes w2 r hV UHpp ()t . (8.2.9) 0 wt 2 e r Forpp e , the relation (8.2.9) yields Vt 0 , and then from (8.2.4) we have ur 0 , and rr 0 . Denoting by h0 , H0 the equilibrium values of the thickness of the wall and of the effective inertial thickness, the conditions for the conservation of mass of the wall and the tissue are given by UT00rHzdd UT r0 d Hz00 d, UTrhzdd UT 0 r0 dhz0d , (8.2.10) or rH r00 H , rh r00 h . (8.2.11) Taking into account that 224 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 2 A S(ru0 r ) , (8.2.12) the equation (8.2.2) becomes wwuuur wv rr0 (1 ) v r 0 . (8.2.13) wwwtrzz2 0 Substituting (8.2.4) and (8.2.11) into (8.2.9) the equation of motion of the wall becomes ur 2 ha0 (1 ) w uurrpp e r0 U0 2 (1 ) Eur . (8.2.14) wt Hr00 2 ur Hr00(1 ) r0 Let us introduce the dimensionless independent variables zLz 0 c , tT 0tc , (8.2.15) with 2 rH000U rH000U L0 , T0 , (8.2.16) 2U hE0 and dimensionless dependent variables vp, and J vz cv0 , ppe pp0 , urz 0 J , (8.2.17) wherec0 andp0 are given by Lh00E hE0 c0 , p0 . (8.2.18) Tr002U 2r0 Therefore, the basic equations (8.2.1), (8.2.13) and (8.2.14) become wwwvvp v 0 , (8.2.19) wwwtzzccc wJ 1 wv wJ J(1 ) v 0 , (8.2.20) wwwtzcc2 zc wJ2 1(1J Ja) p(1J ) . (8.2.21) wtc2 2(1J) At the end of the diastole period, the flow velocity vz and the acceleration of tube wall vanish. The diastolic pressure equals the lowest blood pressure. The quantities J 0 , p0 can be related by (8.2.21), if we write pppp00e 0 and ()urr 00 J 0 , that is 2(1J0 Ja 0 ) p0 2 . (8.2.22) (1J 0 ) INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 225 Now, let us study the asymptotic behavior of the governing equations (8.2.19)– (8.2.21). To describe the nonlinear asymptotic behavior, Gardner and Morikawa introduced a scale transformation to combine it with a perturbation expansion of the dependent variable. Employing this method, we firstly linearise the equations in the form wwvp 0 , (8.2.23) wwtzcc wJ 1 wv J(1 ) 0 , (8.2.24) wwtzcc2 0 2 (1 (2a 1) ) wJ 1 J J0 22 p(1J 0 ) , (8.2.25) wtc 2(1J 0 ) wherep and J are defined as pp 0 p , J J0 J . (8.2.26) Suppose we have harmonic solutions vp,, J , with exponential factor of the form exp[i(kzccZ t )] . (8.2.27) In this case, we obtain a system of homogeneous equations in amplitudes vp,, J . The dispersion relations are obtained by requiring that the determinant of the system vanishes gk 1(21)Ja Z , g 0 . (8.2.28) 2 1 k (1J 0 ) We expand Z()k into a Taylor series about the origin for small k , and retain terms up to the third power of k . So, we have k 2 Z|()tgk (1 ), (8.2.29) 2 and the exponential (8.2.27) becomes kgt3 c exp[i{kz (cc gt ) }] . (8.2.30) 2 The similarity of asymptotic behavior holds for a coordinate transformation which satisfies zgtcc const. (8.2.31) ()gtc 1/3 So, the scale transformation, for which the invariance of (8.2.31) hold, can be defined as 226 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS [ H()zgtcc, W H H gtc , (8.2.32) with H a small parameter. The perturbation expansions of vp,, J with respect to H , are given by f f f vvn (,), pp n (,), n (,). (8.2.33) ¦Hn [W ¦Hn [WJ ¦HJ[Wn n 1 n 1 n 1 The dependent variables vp,, J depend on [ and W , and the nonlinear equations (8.2.19)–(8.2.21), written with respect to [ and W , are ww wwvp gvv()H 0 , (8.2.34) w[ wW w[ w[ ww1 wwJv gv()(1)HJJJ 0 , (8.2.35) w[ wW 2 0 w[ w[ www2221 gp223(2H H H )()(1 J p J J) w[22w[wW wW 2 00 (8.2.36) ()[1()]JJ a JJ 00 0. (1J0 J ) Substitute (8.2.33) into (8.2.34)–(8.2.36) and equate the coefficients of like powers ofH . The terms that are proportional to H0 , give equation (8.2.22), and the terms proportional to H give wwvp g 11 0 , w[ w[ wJ 1 wv Jg 1 (1 )1 0 , (8.2.37) w[2 0 w[ 2 11 (1JJJ 2aa001 ) JJpp 01(1 0 ) 1 2 0 . 22 (1)J 0 Also, the terms proportional to H2 give wwwwvvvp ggv2112 0 , w[ wW1 w[ w[ wJ wJ 11wvv w wJ JJgg21(1 ) 2 11v 0 , (8.2.38) w[ wW 22011w[ w[ w[ INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 227 wJ2 11 1 gp2 1 JJJ(1 ) pp w[2 2202 0 2 2 11 2 2 (1JJJ 2aa002 ) (1)a J1 23 0. (1J00 ) (1J ) From (8.2.37) we obtain by integrating 1 vpf W(), (8.2.39) 11g 2g v11 JMW(), (8.2.40) 1J 0 2g 2 p1 J1 , (8.2.41) 1 J 0 where the functionsf ()W and MW()can be determined from initial conditions. We can have in particular f ()W MW () 0, (8.2.42) for pp 0 , J J0 , for vz 0 , (8.2.43) or p1 0 , J1 0 , for v1 0 . (8.2.44) In these conditions, (8.2.39) and (8.2.40) give again (8.2.41). The quantities v11, p andJ1 are related between themselves by p1 2g v1 J1 . (8.2.45) g 1J 0 Eliminating vp2 , 2 and J2 from (8.2.38) and (8.2.45), we obtain the KdV equations wwwvvv1 3 111Kv 0 , (8.2.46) wW1 w[ 2 w[3 wwwppp1 3 111Lp 0 , (8.2.47) wW1 w[ 2 w[3 wJ wJ1 w3 J 111JM 0 , (8.2.48) wW1 w[ 2 w[3 228 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS where the constants K ,L and M are given by (1J 0 )[(1 2aa ) 3(2 1) J 0 ] K 3/2 , (8.2.49) 4[1J (2a 1) 0 ] K 2g L , M K . (8.2.50) g 1J 0 8.3 Two-soliton solutions The flow velocity equation (8.2.46) may be written as UUUUW 6 x xxx 0 , (8.3.1) by the transformations 3 vU , [ X , W 2T . (8.3.2) 1 K The equation (8.3.1) admits a soliton solution given by Uk 222 sech [ kXkT (G 4 2 ) ], (8.3.3) with G const. The solution for flow velocity is obtained from (8.2.17) and (8.2.33) 6 22k vztz ( , ) ck0 sech [ ( zVt )G ] , (8.3.4) KL0 where 22 2 Hkk , Vg ck0 (1 2 ) . (8.3.5) The equations (8.2.47), (8.2.48) can be solved in a similar way, and the solutions for fluid pressure and radial displacement are given by 6 22k pzt(,) p00 pksech [ ( zG Vt ) ] , (8.3.6) LL0 6 22k uztr (,) r00J rk 0 sech [ ( zVt ) G ] . (8.3.7) ML0 The soliton solutions describe the pulsatile pulses in which the amplitude and velocity are related. The pulses with large amplitude are narrow in width and move rapidly. This feature can explain that the steepening in the arterial pulse occurs in accordance with the increase in the velocity pulse. For arbitrary initial conditions the solutions UX(,)T of (8.3.1) are obtained exactly by applying the inverse scattering transform. As T of, the solution approaches asymptotically to N soliton solutions. The amplitudes and the velocities of these INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 229 2 solutions are determined by the eigenvalues kn of N bound states of the Schrödinger equation for which the potential is the initial condition UX(,0). Consider now the arbitrary initial conditions for the KdV equation (8.3.1) which can 2 2 support two bound states at k1 and k2 KK K UX( ,0) v1 ( [ ,0) vz (z ,0) vz( ,0) , 33HH3 c0 1 X [ Hzc H z . (8.3.8) L0 The solution is obtained with no reflection, Rk(,0) 0 , and yields as T of to a two-soliton solution given by UXT( , ) 2 k22 sech [(k X 4k 2T)]G 111 1 (8.3.9) 22 2 2kkk222 sech [ (X 4 T ) G2 ]. The constants G1 ,G2 are arbitrary. The flow velocity is described as a two-soliton solution and can be regarded as being composed from a main pulse and an associated dicrotic wave. From (8.3.9) it results the solution for the flow velocity of blood 6 k vzt( , ) ck22 sech[ 1 (z 4kt2 )]G z KL01 11 0 (8.3.10) 6 22k2 2 ck02sech [(z 4kt2 ) G2 ]. KL0 In a similar way we obtain two-soliton solutions for blood pressure and for radial displacement 6 k pzt(,) p pk22sech [ 1 (z 4)kt2 G] 00LL1 1 1 0 (8.3.11) 6 22k2 2 pk02sech [(z 4kt2 ) G2 ], LL0 6 k uzt(,) uJ rk22sech [ 1 (4zk 2t)]G r 00 ML01 1 1 0 (8.3.12) 6 22k2 2 rk02sech[ ( z 4kt2 ) G2]. ML0 The time t0 required for a pulse starting from the heart to cause the steepening in the aorta may be estimated by considering the equation (8.2.46) in which the third term is discarded 230 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS wwvv 11Kv 0 . (8.3.13) wW1 w[ The solution of (8.3.13) can be regarded as a functional relation vf1 ([WKv1 ) . (8.3.14) Therefore, the scaled time t0 , which causes the steepening, may be connected with a scaled initial pulse of width [0 , on the [ -axis, by an approximate relation 1 W#[Kv() , (8.3.15) 01m 2 0 where()v1 m is the maximum value of v1 . Inserting the transform relations obtained from (8.2.15), (8.2.17), (8.2.32) and (8.2.33) 1 [00 HzLc H d0 , 1 W000 HHtTc HH t0 , (8.3.16) 111 vvcv10mm H() H (zm ) , into (8.3.15) we have d0 t0 # , (8.3.17) 2()Kg vz m wheredz0 is the width of the initial pulse on the -axis. Taking K =1, g=1 and ()vzm 0.5m/s, d0 0.3 m, from the experimental data, the time t0 is estimated to be about 0.3 s. For V 5 m/s, the distance that the wave needs to travel until the soliton is formed is estimated to be about 1.5 m. But the distance from the heart to the abdominal aorta is about 0.5 m. Therefore, in the abdominal aorta the soliton is not yet formed. The steepening phenomenon can be interpreted as the generating and growing of solitons in large arteries. The condition for an equilibrium state of the system is given by pp 0 , vz 0 , for all z , (8.3.18) and the boundary conditions of the system are pp 0 , vz 0 , as z rf. (8.3.19) From (8.2.22), (8.2.28) and (8.3.6) we have a 3 pp0 e 1 cc#0{1 ( ) } , (8.3.20) 2 24 p0 1 k aL3 pp0 em p p0 Vc#0{1 ( ) } {1 } . (8.3.21) 24pp00 3 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 231 The velocity of sound wave with small amplitude near the equilibrium state is given 21/2 by (8.2.28), that is gc0 (1 k ) , and the phase velocity of a soliton with large 2 amplitude, which satisfy the boundary condition (8.3.19), is given as gck0 (1 2 ) . These velocities depend on g and J 0 . But J 0 and p 0 are related by (8.2.22), and thus theses velocities change with the lowest pressure p0 . The phase velocity includes a term of k2 which is proportional to the amplitude of the pulse. Then the efficiency of blood circulation depends on the lowest blood pressure p0 and on the difference between the highest and the lowest pressure of blood pm p0 . In this section, the results obtained from this theory are compared with the experimental measurements given by McDonald for the dog. These experimental results are summarized as follow (Yomosa): – for the thoracic aorta 3 h0 9 3 r0 510u m, 0.12 , U 1.05u 10 kg/m , r0 9 35 U0 1.06 u 10 kg/m , E 5.49u 10 Pa, (8.3.22) vm 0.55 m/s, ppme 102 mmHg=102u133.3 Pa, pp0 e 81mmHg = 81u133.3 Pa, pm 21mmHg = 21u 133.3 Pa, urrm 0.05 0 , v 5.5 m/s, f 3.65 cycles/s, wheref is frequency. – for the femoral artery 3 h0 9 3 r0 1.5u 10 m, 0.12 , U 1.05 u 10 kg/m , r0 9 35 U0 1.06 u 10 kg/m , E 14.1u 10 Pa, (8.3.23) vm 0.4 m/s, ppme 110 mmHg = 110u133.3 Pa, pp0 e 78 mmHg = 78u133.3 Pa, pm 32 mmHg = 32u133.3 Pa , urrm 0.03 0 , v 1 m/s, f 3.65 cycles/s. In the static case we have h0 J(1Ja ) ppe E 2 . (8.3.24) r0 (1 J ) r Figure 8.3.1 represents an experimental diagram ( ppe ,1 J ) obtained by r0 McDonald for the dog. From this graph and (8.3.24) we estimate E 5.37u105 Pa and h a 1.95 for the thoracic aorta, by using 0 0.12 . The Young’s modulus thus r0 estimated is in good agreement with (8.3.22). 232 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS r Figure 8.3.1 Experimental diagram ( ppe ,1 J ) for dog (McDonald). r0 For the femoral artery the same value for the nonlinear elastic coefficient a is h0 considered. The parameters c0 and p0 are estimated by substituting the values of , r0 U andE given by (8.3.22) and (8.3.23) into (8.2.18) c0 5.6m/s ,p0 247 mmHg = 247u 133.3 Pa, (thoracic aorta) , (8.3.25) c0 8.98m/s ,p0 635 mmHg = 635u 133.3 Pa, (femoral artery). (8.3.26) The values of p 0 which correspond to the lowest pressures pp0 e 81mmHg in the thoracic aorta, and pp0 e 78 mmHg in the femoral artery are determined from (8.2.17) and (8.3.25), (8.3.26). p 0 0.328 , (thoracic aorta), (8.3.27) p0 0.123 . (femoral artery). (8.3.28) The value of J 0 is calculated from (8.2.22) and (8.3.27), (8.3.28) J 0 = 0.169, (thoracic aorta), (8.3.29) J 0 = 0.062, (femoral artery). (8.3.30) The values of g,,KL and M are estimated from (8.2.28), (8.2.49) and (8.2.50) g 1.044 , K 1.024 , L 0.980 , M 1.829 , (thoracic aorta), (8.3.31) g 1.023 , K 1.127 , L 1.102 , M 2.171, (femoral artery). (8.3.32) From the soliton solutions (8.3.4), (8.3.6) and (8.3.7) we derive 6 6 6 vck 2 , pp k2 , ()u rk2 , (8.3.33) m K 0 m L 0 rm M 0 Substituting into (8.3.33) the experimental values of vm , pm and (urm) given by (8.3.22) and (8.3.23), we estimate three values of k for the soliton in the thoracic aorta INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 233 k 0.129, k 0.118, k 0.123. (8.3.34) In a similar way, three values of k are obtained for the femoral artery k 0.091, k 0.096, k 0.104. (8.3.35) Therefore, we can take k # 0.12 for the thoracic aorta, and k # 0.09 for the femoral artery. From Figure 8.2.1 we can estimate the amplitudes of the dicrotic flow and d d pressure waves in the femoral artery, denoted by vm and pm d d vm = 0.05 m/s, pm = 5 mmHg . (8.3.36) Then we estimate the value ofk for the dicrotic wave in the femoral artery for the first two equations (8.3.33) kd =0.032,kd =0.038. (8.3.37) Therefore we can regard the wave in the femoral artery as a two-soliton wave which is characterized by k1 # 0.09 and k2 # 0.035. Substituting the values of g,c0 and k estimated above, into (8.3.5) 2 we have the wave velocities V # 6.02 m/s, (thoracic aorta), V # 9.34 m/s, (femoral artery). (8.3.38) These values are reasonable compared with the experimental data (8.3.22) and (8.3.23). Since all values of k estimated until now satisfy k2 1 , we can see from (8.3.5) and (8.2.28) that the solitons velocities in arteries nearly equal the sound wave velocities in them. From the profiles of the flow and pressure waves in Figure 8.2.1, we can estimate the widths of theses pulses. We have 2L 1 V 0 # 0.11m , (thoracic aorta), k 15 f 2L 1 V 0 # 0.0856m , (femoral artery), (8.3.39) k 30 f and 2L 0 # 0.02s , (thoracic aorta), kV 2L 0 # 0.01s , (femoral artery). (8.3.40) kV 234 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS L0 The characteristic lengths L0 and time T0 are estimated by inserting the k c0 values into (8.3.39) and by using (8.3.25) and (8.3.26) 2 3 L0 0.66u 10 m , T0 1.2u 10 s , (thoracic aorta), 2 3 L0 0.38u 10 m , T0 0.4u 10 s . (femoral artery). (8.3.41) We can estimate the values of effective inertial thickness of the wall by using (8.2.16) and (8.3.22) and (8.3.41) 2 H0 1.73 10 m , (thoracic aorta), 2 H 0 1.93u 10 m , (femoral artery). (8.3.42) The additional inertial thickness due to the mass of the tissue hHh00c 0 is estimated as 2 h0c 1.7u 10 m , (thoracic aorta), 2 h0c 1.9u 10 m , (femoral artery). (8.3.43) Figure 8.3.2 Two-soliton solution U(0,t ) in the femoral artery of the dog. Figure 8.3.2 represents the two-soliton solution U (0,t ) given by (8.3.9) for the femoral artery of the dog, with 2 L0 0.38u 10 m , k1 0.09 , k2 0.035 , V 9.34m/s , G1 0 , G2 2.9 . The frequency of generating the solitons is 3.65 cycles/s, the period is Tf 1/ 0.27s , and the wavelength is TV 2.56m . For the thoracic aorta we have Tf 1/ 0.27s , V 6.02 m/s , TV 1.64m . INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 235 Figure 8.3.3 Two-soliton solution of flow velocity in femoral artery of the dog. Figure 8.3.3 represents the time variation of the two-soliton solution of the flow velocity in the femoral artery of the dog, in an arbitrary location, for k1 0.09 , k2 0.035 ,,V 9.34m/s K 1.127 , c0 8.98m/s , G1 1.7 ,.G2 1.3 The frequency is 3.65 cycles/s. Figure 8.3.4 represents the time variation of the two-soliton solution of the blood 2 pressure in the femoral artery of the dog, in an arbitrary location, for L0 0.38u 10 m , k1 0.096 , k2 0.041 ,,V 9.34m/s L 1.102 , p0 635mmHg 635u 133.3Pa , G1 0 . Figure 8.3.4 Two-soliton solution of blood pressure in the femoral artery of the dog. By comparing these results with the experimental results given by McDonald we observe that waves are narrower than in experimental data, which can be explained by the fact that the viscosity is completely neglected. It is interesting to note that the features of the graphs (8.3.3) and (8.3.4) are very similar to the Hamiltonian of the sine- Gordon equation behavior (Hoenselaers and Micciché). 8.4 A micropolar model of blood flow in arteries The blood is a rheologically complex fluid being a suspension of particles (red and white cells, platelets) undergoing unsteady flow through vessels. Eringen introduced a mathematical model for such fluids, called micropolar fluids, in 1966. This theory 236 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS exhibits microrotational effects such as those experienced by blood in the larger vessels of the circulation for the dog (McDonald). Ariman and his coworkers, Turk and Sylvester, Easwaran and Majumdar, have applied the Eringen theory for describing the time-dependent blood flow. Their model formulates a new kinematic variable called microrotation describing the individual rotation of particles within the continuum, independent of the velocity field. In this theory an arbitrary pressure gradient as a sum of sine functions was considered. But the pulsatile character of the blood flow suggests the using the soliton theory. Let us consider the thin-walled elastic tube is infinitely long, straight, circular and homogeneous, embedded in the tissue and filled with the blood Consider the one-dimensional fields of longitudinal flow velocity vx(,)t , the micro- gyration wxt(,) , the blood pressure pxt(,) and the cross-sectional area A(,)xt under the assumption of uniform distributions of v and p over the cross-section of the tube. Here x is the axial distance along the vessel and t , the time. The radial component of the flow velocity is neglected in comparison with the axial component. The dimensional governing equations of the micropolar fluid dynamics are given by wwvv11 w p wDw2 v w v DP() 0 , (8.4.1) wwUwUtx x wx2 Uwx wDwJwDwv2 w j 2 w 0 , (8.4.2) wUwUtxwx2 U wwAvA() 0 , (8.4.3) wwtx whereU is the fluid density, DPJ,, the blood coefficients and j the microinertia 2J coefficient ( j ) . The coefficient D depends on the cellular concentration 2PD (hematocrit) and is determined from the equation D222 D(2 PJqq ) J P 0 , (8.4.4) whereq has been found experimentally by Bugliarello and Sevilla (see Ariman et al.) given as q 105-1 m ,P represents the viscosity of blood plasma (0.02 poise at 300 C ). 1 The ratio is defining the red cell diameter for blood. q The equation of radial motion of the arterial wall is given by (8.2.14) u 2 ha0 (1 ) w uupp e r0 U0 2 (1 ) Eu , (8.4.5) wt Hr00 2 u Hr00(1 ) r0 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 237 with U0 the density of the material of the wall, rxt(,) the radius of the tube, h the thickness of the wall in radial direction, pe the pressure outside the tube which can be regarded as about the same as the atmospheric pressure, V the extending stress in the tangential direction. The density of the materials of the wall and the tissue are assumed to be equal. In (8.4.5) u is the radial displacement of the wall, E the Young's modulus of the elasticity, a a nonlinear coefficient of elasticity. 2 Taking account of A S()ru0 , we obtain from (8.4.5) the motion equation of the cross-sectional area 2 wwAA112 2 () fAp (,), (8.4.6) wt 2AtwU0 where ppSSe 2[ EhraA00 (/)] r0 fAp(,) 2 A 2 (/)AS r0 . (8.4.7) Hr00 Hr00 We introduce dimensionless, noted by prime, variables rh00UU 01/2 r0 01/2 l0 x ltcx000 (),(),, c , 2U Etl00 tvxt(,) pxt(,) pe tvxtpxtcc ,(,) , c (,) 2 , (8.4.8) tc00Uc 0 A(,)xt 22r wtwAxtcc 00,(,) , A S rA0 , S rr ,c . A00r The dimensionless equations can be written by dropping the prime as wwwwvvp2 vw w vmn 0 , (8.4.9) wwwtxxwx2 w x wwwwv2 w bqg dw 0 , (8.4.10) wwtxwx2 wwAvA() 0 , (8.4.11) wwtx ww2 AA1 ()2 fAp (,) , (8.4.12) wt 2 2Atw where 238 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS DP D j mnb 22,,, UUct00 cl 00 l 0 D JDtt00 qgd 242,,, UUUct00 l0 l0 (8.4.13) ppSe 2[ EhraA00 (/ r0 )] fAp(,) 2 A4 ( A / S r0 ), Hr00 Hr00 2 2Uc0 2(2aa 1) 2 2(1 a ) bb12 ,,,. bb 34 hE r00 r r 0 Let us assume an unsteady arbitrary pressure gradient of the form (Freeman) 2 pxt A2 bˆ kx t , (8.4.14) ( , ) ¦ ii sech (Z ) 1 ˆ with Aii,,bk unknown constants and Z the circular frequency. The governing flow equations (8.4.12)–(8.4.15) are solved in the condition of satisfying (8.4.16) and the initial conditions wv vv(0,0) , (0,0) vww , (0,0) , 01t 0 w (8.4.15) wwwA (0,0) wA , (0,0) a , (0,0) a . wwtt10 1 The scenario for variation of coefficient D gives the effect of the volume concentration of the red cell upon the flow behavior of blood. Equations (8.4.13)–(8.4.16) can be simplified into a form suitable for obtaining the periodic wave solutions. It will be of interest to obtain the wave of permanent form solutions of this system using the change of the variable Z ykxctc (), , (8.4.16) k wherec is the unknown wave velocity. In order to simplify equations we consider a 1 . The system of equations are written as cvccc vv p mkv ccc nw 0, bckw c qkv c gk2 w cc dw 0, 1 (8.4.17) cAcc() vA 0, k22 c ( A ccc A 2 ) f (,). A p 2A The profiles of the flow velocity, the blood pressure and the cross-sectional area are numerically calculated using the data given by (8.3.22) for the thoracic aorta of the dog. INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 239 Figure 8.4.1 Variation of the blood pressure in one period. Figure 8.4.2 Variation of the flow velocity in one period. For the blood coefficients we choose P 0.02poise , J 0.6 mkg/s and j 20m2 . From (8.4.4) we obtain D 0.02poise . By equating the width of the blood pressure two-soliton wave to the width (the wavelength is 162u102 m ) of the pulse wave observed experimentally for the thoracic aorta of the dog, we obtain T 0.27s and c 6m/s. The dimensionless initial conditions are given by vv(0) 0.5,c (0) 0, ww(0) 0.2,c (0) 0, AA(0) 1,c (0) 0. ˆˆ Constants from (8.4.18) are found as AA12 1 , 0.6 , b 1 2.3 , b 2 4.5 . Figures 8.5.1–8.5.3 are plots of the blood pressure, the longitudinal flow velocity and the microgyration for one period. It is seen that the systolic and diastolic peaks exist in all graphs. The systolic and diastolic pressure peaks are related to the flow velocity peaks. The phase portraits for flow velocity and the microgyration are plotted in Figures 8.4.4–8.4.5 after four periods. For an increasing number of periods the graphs remain unchangeable. The motion becomes stable, the role of the initial conditions being very small. In Figure 8.4.6 the cross-sectional area is plotted for one period. 240 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS Figure 8.4.3 Variation of the microgyration in one period. Figure 8.4.4 Phase portrait for the flow velocity in one period. To illustrate the effect of the hematocrit on the flow velocity and on the micro- gyration we consider four values for the coefficient D . Figures 8.4.7–8.4.8 show a decreasing of the amplitude of the flow velocity and of the microgyration for increasing hematocrit. The results are reasonable due to the fact that the increase of viscosity overshadows the inertial effects leading to a reduction of the amplitudes and are in agreement with the experiments. Figure 8.4.5 Phase portrait for microgyration in one period. INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS 241 Figure 8.4.6 Variation of the area in one period. Figure 8.4.7 Effect of hematocrit on flow velocity. Figure 8.4.8 Effect of hematocrit on microgyration. The two-soliton representation for the blood pressure may be viewed as an interaction of two pulses. They are superimposed but retain their identity and do not destroy each other into the flow. This assures the stability of the blood circulation. The flow velocity, the microgyration and the cross-sectional are calculated as functions of the two-soliton blood pressure pulse. The phase portraits demonstrate the stability of the blood motion (Tabor). Chapter 9 INTERMODAL INTERACTION OF WAVES 9.1 Scope of the chapter The modal interaction in a Toda lattice was discussed in section 6.5. In this chapter we discuss other phenomena by means of intermodal interaction of waves. In the first four sections we explain the subharmonic generation of waves in piezoelectric plates with Cantor-like structure. We show that subharmonic generation of waves is due to a nonlinear superposition of cnoidal waves in both phonon and fracton vibration regimes. Alippi (1982) and Alippi et al. (1988) and Crăciun et al. (1992) have proved experimentally the evidence of extremely low thresholds for subharmonic generation of ultrasonic waves in one-dimensional artificial piezoelectric plates with Cantor-like structure, as compared to the corresponding homogeneous and periodical plates. An anharmonic coupling between the extended-vibration (phonon) and the localized-mode (fracton) regimes explained this phenomenon. They demonstrate that the large enhancement of nonlinear interaction results from the more favorable frequency and spatial matching of coupled modes (fractons and phonons) in Cantor-like structure. Section 9.5 presents the Yih analysis of the interaction of internal solitary waves of different modes in an incompressible fluid with an exponential stratification in densities. The turbulent flow of a micropolar fluid downwards on an inclined open channel is studied in sections 9.6 and 9.7. The wave profile moves downstream as a linear superposition of soliton waves at a constant speed and without distortion. In a micropolar fluid the motion is described not only by a deformation but also by a microrotation giving six degrees of freedom (Eringen 1966, Brulin 1982). The interaction between parts of the fluid is transmitted not only by a force but also by a torque, resulting in asymmetric stresses and couple stresses. The micropolar theory is employed here to obtain solutions, which are periodic with respect to distance, describing the phenomenon called roll-waves for water flow along a wide inclined channel. The soliton representation is not so surprising in this case because Dressler in 1949 studying the roll-waves motion of the shallow water in inclined open channels found an equivalent form expressed as a cnoidal wave for a flow subject to the Chezy turbulent resisting force. The last section presents the results of Shinbrot concerning the effect of surface tension on the solitary waves. The chapter refers to the works by Munteanu and Donescu (2002), Chiroiu et al. (2001b), Yih (1994) and Shinbrot (1981). INTERMODAL INTERACTION OF WAVES 243 9.2 A plate with Cantor-like structure We consider a composite plate formed by alternating elements of nonlinear isotropic piezoelectric ceramics (PZ) and a nonlinear isotropic epoxy resin (ER), following a triadic Cantor sequence (Figure 9.2.1). Crăciun, Alippi and coworkers constructed an artificial one-dimensional Cantor structure. The Cantor set is an elementary example of a fractal. The Cantor set is generated by iteration of a single operation on a line of unit length. The operation consists of removing the middle third from each line segment of the previous set. As the number of iterations increases, the number of separate pieces tends to infinity, but the length of each one approaches zero. The property of invariance under a change of scale is called self-similarity and is common to many fractals. In contrast to a line with its infinite number of points and finite length, the Cantor set has an infinite number of points but zero length. The dimension of the Cantor set is less than 1. Denoting byN the number of segments of length H , the dimension of the Cantor set is given by Baker and Gollub log N log 2n log 2 d lim lim 1. Ho0 log(1/H ) nof log 3n log 3 We consider the same sample using a triadic Cantor sequence up to the fourth generation (31 elements). A rectangular coordinate system Ox123 x x is employed. The origin of the coordinate system is located at the left end, in the middle plane of the sample, with the axis Ox1 in-plane and normal to the layers and Ox3 out-plane, normal to the plate. The length of the plate is l , the width of the smallest layer is l /81 and the thickness of the plate is h . The width of the plate is d . Let the regions occupied by the plate be V VV ep where V p and V e are the regions occupied by PZ and ER layers. The boundary surface of V be S partitioned in the following way pe pe SS 11 S S21, SSS 1 2 0 , where p p Sxh13 {/2,0 r x1l} , is the boundary surface of V , e e Sxh13 {/2,0 r x1l}, is the boundary surface of V , Sx21 {0,,/2 xlhxh 1 dd3 /2} . pe Let the unit outward normal of S be ni the interfaces between constituents be I . An index followed by a comma represents partial differentiation with respect to space variables and a superposed dot indicates differentiation with respect to time. Throughout the present paper repeated indices denote summation over the range (1,2,3). In order to investigate theoretically the existence of multiple fracton and multiple phonon mode regimes in the displacement field for a piezoelectric plate with Cantor- like structure it is customary to consider: first, the nonlinear geometrical relations between the components of deformation and those of the displacement vector; second, to retain in the constitutive equations besides the linear terms also the nonlinear terms of lowest order. So, we have considered the piezoelectric material to be nonlinear and 244 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS isotropic, characterized by two second-order elastic constants, three third-order elastic constants, two (linear and nonlinear) dielectric constants and two (linear and nonlinear) coefficients of piezoelectricity. In our first attempt to investigate the existence of multiple fracton and multiple phonon mode regimes, we considered the case of anisotropic piezoelectric material with monoclinic symmetry and we neglected the third-order constants. In spite of the bigger number of elastic, dielectric and piezoelectric constants, the results did not show clearly the existence of localized and extended modes regimes. In conclusion, a quantitative knowledge of the second-order material constants is essential for the analysis of the fracton and phonon mode regimes for a piezoelectric plate (ùoós). The dashed regions are occupied by piezoelectric p p ceramic of total volume V and boundary external surface S1 . The white regions are e e occupied by epoxy resin of total volume V and boundary external surface S1 . Figure 9.2.1 The plate with Cantor-like structure. pe The lateral surfaces are S2 and the interfaces between constituents I . In the following, the basic equations are written for piezoelectric and non-piezoelectric materials: a) Piezoelectric material (PZ). 1. The quasistatic motion equations p p Uututiijkikj (),,j , in V , (9.2.1) p Dii, 0 , EiiM, 0 , in V , (9.2.2) where U p is the density, ui is the displacement vector, tij is the stress tensor, Di is the electric induction vector, Ei is the electric field and M is the electric potential. 2. The constitutive equations ppppp2 tAij OHGPH kk ij 23ij HH il jl B HHkk ijC HG kk ij in V p , (9.2.3) pp p p GHGHGHeEk k ij eE k k ll ij eE k k ll ij eE k k ij , INTERMODAL INTERACTION OF WAVES 245 111 DE HHHHHpppp Eee22 e p2 , in V p , (9.2.4) iiiikkikki222 kl 1 H (uuuu ) , in V , (9.2.5) ij2 i,,,, j ji li l j p p p pp whereHij is the strain tensor, OP, are the Lam- constants, A ,,BCare the Landau p p p p p constants, HH, i ( H321 H H ) are the linear and nonlinear dielectric constants, p p pp p p p p p p pp ei ()ee123 e , ei ()ee123 e and ei ( ee12 e3 ) are the linear and 2222 nonlinear coefficients of piezoelectricity and EEE 12E3 . 3. The boundary conditions p tnij j Tn ij j T i , on S1 , (9.2.6) p Dnii d, M M, on S1 , (9.2.7) where Ti , d , M are quantities prescribed on the boundary and Tij is the Maxwell stress 0 tensor. We consider that a periodical electric field EEii exp(iZ t ) is applied to both surfaces of the plate to excite the Lamb waves, over a wide frequency range (10MHz Z / 2 S 5MHz ). The action of this field is described by Maxwell stress tensor Tij (Kapelewski and Michalec) 11 TE ( EEG2 ) , on S p . (9.2.8) ij 42S ij ij 1 p The boundary conditions (9.2.6)–(9.2.7) on S1 are rewritten as 11 tEEtEE ,( 22 ) , on S p , (9.2.9) 13 48SS1 3 33 3 1 1 p DEEE3311 ,, on S1 . (9.2.10) b) Non-piezoelectric material (ER). 4. The motion equations e e Uutiij , j , in V . (9.2.11) 5. The constitutive equations eeeee2 e tAij OHGPH kk ij 23ij HH il jl B HHkk ijC HG kk ij , in V . (9.2.12) 6. The boundary conditions e tnij j 0 , on S1 , (9.2.13) 246 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS or e tt13 31 0 , on S1 . (9.2.14) 7. The boundary conditions on S2 uu13 0 , on S2 . (9.2.15) 8. The conditions on interfaces between constituents I pe . At the interfaces between constituents the displacement and the traction vectors are continuous. pe []uu13 []0 , []tt11 []0 13 , on I , (9.2.16) where the bracket indicates a jump across the interface and k 1, 3 . The 3D problem can be reduced to a 2D problem, if we consider all quantities independent with respect to x2 , and u2 0 , E2 0 . We have u1113 uxxt(, ,), u3313 uxxt(, ,), in V , (9.2.17) p E1, M13 , E M ,3 , M M(,x13xt ,), in V . (9.2.18) We express the elastic potentials I and \ u1,1, I \3 , u3,3,1 I \ , in V , (9.2.19) in the form I [A cos( DxB33 ) sin( D x )] I ( xt1 , ) , p \ [cos()CxDxx E33 sin()](,) E \ 1 t , in V , (9.2.20) M [ExFxx cos( J33 ) sin( J )] M (1 ,t ) , and ˆ ˆ I [A cos( DˆˆxB33 ) sin( D x )] I ( xt1 , ) , ˆˆˆ ˆ e \ [cos()CxDxx E33 sin()](,) E \ 1 t , in V , (9.2.21) whereI\, and M are unknown functions. Kapelewski and Michalec obtained the analytical proof of the existence of solitary surface waves for a free nonlinear semi-infinite isotropic piezoelectric medium. Following this work we have observed that the governing equations (9.2.1)–(9.2.2) and (9.2.11) can support for I\, , M , Iˆ,\ˆ and Mˆ certain particular functions I\, and M having the form INTERMODAL INTERACTION OF WAVES 247 M sh(K ) I Isech(K ) , M 0 \2 , \ (\ 1)sech(K )exp( ) 1 , (9.2.22) 0 2 0 \ 0 I 0 where K x1 vt ,v the velocity of wave. This result plays an essential role in our strategy to determine the functions I,\ and M by using the cnoidal method. Using the cnoidal representations, the displacement field in V p and V e is given by uA [ cos(DDI4 xB ) sin( x )] [log ]ccc 1330n (9.2.23) E[sin()cos()][log],CxDx E330 E E \ 4n cc uAxBx [D sin( D ) D cos( D )] I [log 4 ]cc 3330n (9.2.24) EE\4[CxDx cos(33 ) sin( )] 0 [logn ]ccc , in V p , and uA [ˆ cos(DDI4ˆˆ xB )ˆ sin( x )] [log ]ccc 1330n (9.2.25) ˆˆ ˆ ˆˆ ˆ E[sin()cos()][log],CxDx E33 E E \ 0 4n cc uAxBx [Dˆˆˆ sin( D ) ˆ D ˆˆ cos( D )] I [log 4 ]cc 3330n (9.2.26) ˆˆˆ ˆ EE\4[CxDx cos(33 ) sin( )] 0 [logn ]ccc , in V e . The electric field in V p is given by EExFx133 [ cos( J ) sin( J ) M 0 [log 4n ]ccc , (9.2.27) EExF33 [sin()J J J cos()[log] J x 3 M 0 4n cc . (9.2.28) In these expressions, the prime means the differentiation with respect to x { x1 . We have used here the following representations for the functions I,\ and M w2 I(xt , ) I4KKK log ( , ,..., ) , 01w2 x nn2 w2 M(xt , ) M4KKK log ( , ,..., ) , (9.2.29) 01w2 x n 2n w2 \(xt , ) \4KKK log ( , ,..., ) , 01w2 x n 2n where x { x1 and I00,\ and M 0 constants. In (9.2.29), 4nn(KK12 , ,..., K ) is theta- function (1.4.14) 248 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS f nN1 4KK( , ,..., K ) exp( iM K MBM) , nn12 ¦¦ jjii2 ¦jj Mi f Ji 1,j 1 1ddin and Kj kxjj ZE t j , 1 ddjn, where k j are the wave numbers, Z j the frequencies and E j the phases. The relations (9.2.29) allow the representations I(,)xt IKIKcn ()int (),,M(,)xt MKMKcn ()int () \(,)xt \cn () K \int () K , (9.2.30) w2 where the first terms result from logG (K ) , and the second terms from w2 x w2 F (,K C ) log(1 ) , in accordance to the theorem 1.4.1. wK2 xG() The solutions (9.2.23)–(9.2.28) must satisfy the set of equations (9.2.1)–(9.2.10) in V p , and the set of equations (9.2.11)–(9.2.16) in V e . 9.3 The eigenvalue problem In classical elasticity, there are two types of variational principles for the free vibration of an elastic body. One is associated with the potential energy, the other with the complementary energy. A variational formulation for the free vibration of a piezoelectric body is given by Yang, which is related to the internal energy. The eigenvalue problem for the resonance of the plate is p p Uututiijkikj (),,j , in V , p Dii, 0 , EiiM, 0 , in V , ppppp2 tAij OHGPH kk ij23 ij HH il jlB HH kk ijC HG kk ij in V p , pp p p GHGHGHeEk k ij eE k k ll ij eE k k ll ij eE k k ij , 111 DE HHHHHpppp Eee22 ep 2 , in V p , iiiikkikki222 kl eeeee2 e tAij OHGPH kk ij23 ij HH il jlB HH kk ijC HG kk ij , in V , p ()tutnij k, i kj j 0 , onS , Dnii 0,M 0 , on S1 , e e Uututiijkikj () ,,j , in V , uu13 0 , onS2 , (9.3.1) pe [][]uu13 0, [][]tuttut11kk ,1 1 13kk ,3 3 0 , on I , INTERMODAL INTERACTION OF WAVES 249 1 H ()uuuu , in V , ij,1, 3 . ij2 i,,,, j ji li l j For the eigenvalue problem (9.3.1) for the resonance of the plate we give a formulation of a variational principle by introducing the Yang functional /H(,utDiijij ,,, M i ) 3H(,utDiijij ,,, M i ) , (9.3.2) *()ui where /H(,utD ,,, M ) [ tuDUDtV MH( , ) H ]d iijij i³ ijij,, ii iji ijij V (9.3.3) MtnuSdd, Dn S ³³ji j i i i p S2 S1 1 *()uuu U dV , (9.3.4) ii³ i V 2 and U the internal energy 1 UU (,HDtED ) [ H ] . (9.3.5) ij i2 ij ij i i For free vibrations, the three displacement functions are expressed as (9.2.23)– (9.2.26) and the components of the electric field by (9.2.27)–(9.2.28). The stationary condition on 3 gives the eigenvalue problem (9.3.1) with the stationary value of Z22. The values of Z are sought corresponding to which nontrivial solutions of (9.3.1) exist. The angular frequency Z is related to he circular frequency f by Z 2 Sf . 9.4 Subharmonic waves generation Given a set of measured frequencies fnn Z/2 S, nN 1,2,..., , of the plate we aim at determining the unknown system parameters ˆˆˆˆ ˆ P { k j , Z j , E j ,,Bij ABC,, ,,DEF,, ABCD,,, , DEJ,, , DEˆ , , ij, 1,2,3, 4,5 }, (9.4.1) necessary to analytically construct the solutions of the set of governing equations. The inverse problem we want to consider is, given the solutions representations (9.2.23)–(9.2.28), to find the 55 parameters P given by (9.4.1) by inversion of the measured natural frequencies of the plate. Because we do not have an experimental set of eigenfrequencies we will use for the inverse problem the set of eigenfrequencies computed from the direct problem (9.3.2)–(9.3.5). To determine the parameters P from the measured natural frequencies data by the nonlinear least-squares optimization technique, an objective function must be chosen that measures the agreement between theoretical and experimental data 250 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS N ()PffP {[e ()]}2 min , (9.4.2) ¦ ii o i 1 e wherefi are the measured i -th eigenfrequency,fi the corresponding model prediction, andNN the number of measurements of frequencies ! K , with K the number of unknown parameters which are given by P (for our case this number is 55). The primary goal of the optimization procedure is the minimization of the objective function ()P , where P are the design variables which make up the solutions of the problem. The signs “ o min ”, “ o max ” mean the minimization or maximization of the objective function with some required precision: suppose six decimal places for the variable values are desirable. To measure the accuracy of the identification of P we introduce an error indicator G to estimate the verification of the governing equations, that is 6 ()P Gopt ¦Gk , (9.4.3) k 1 G {( Uput )22 D ( E M )2 }d Vp , (9.4.4) 1,³ iijj iii,, i V p G () Ueut 2 d Ve , (9.4.5) 2,³ iijj V p 11 G {(tEEt )22 [ ( EEDEEES 2 )]222 ( ) ( ) }dp , (9.4.6) 31³ 31333 3133111 p 48SS S1 G []dttS22 e , (9.4.7) 41333³ 1 e S1 G []duuS22 , (9.4.8) 513³ 2 e S1 G [uuttI2222 ]d pe , (9.4.9) 6131113³ I pe wherePopt is the solution of the inverse algorithm (9.4.2), tij are given by (9.2.3) in p e V and (9.2.12) in V ,Di are given by (9.2.4) and Hij by (9.2.5). We define fitness as follows , m F 0 , ()f e 2 . (9.4.10) 0 ¦ i i 1 As the convergence criterion of iterative computations we use the expression Z to be maximum INTERMODAL INTERACTION OF WAVES 251 1 Z log 0 o max . (9.4.11) 2 10 The quality of the model depends on the maximum value of the function Z and the associated value of G(Popt ) . The integrals (9.4.4)–(9.4.8) can be analytically computed. We use the genetic algorithm described in Chapter 7, in this case the binary vector having 55 genes to represent the real values of parameters P given by (9.4.1). The numerical computation of theta-function is drastic by noting that the number of complex exponentials is (2M 1)n 1 (Osborne). For M 5 , n 5 this number is 161050 and for M 10 ,n 100 this number is | 10132 . We consider here the case of five degrees of freedom solutions n 5 and M 5 (5dM n d 5) . The calculus was carried out for l = 67.5mm and h = 0.3mm. The material constants are shown in Table 9.4.1. The calculus was carried out for l = 67.5mm and h = 0.3mm. The material constants are shown in Table 9.4.1 ( I00 M \ 0 1 ). Table 9.4.1 The material constants for piezoelectric ceramics and epoxy resin (Rogacheva) piezoelectric ceramics epoxy resin O 71.6 GPa 42.31 GPa P 35.8 GPa 3.76 GPa A -2000 GPa 2.8 GPa B -1134 GPa 9.7 GPa C -900 GPa -5.7 GPa H 4.065 nF/m - 2.079 nF/m - H1 -0.218 nm/V e1 - -0.435 nm/V - ee1 U 7650 Kg/m 3 1170Kg/m 3 Table 9.4.2 Estimation results: computed eigenfrequencies ZSn /2 100.2 167 217.1 250.5 334 367.4 417.5 501 584.5 r 0.05 r 0.01 r 0.03 r 0.1 r 0.01 r 0.01 r 0.1 r 0.02 r 0.03 617.9 668 835 935.2 1085.5 1169 1269.2 1503 1670 r 0.01 r 0.03 r 0.06 r 0.06 r 0.1 r 0.07 r 0.02 r 0.05 r 0.4 1770.2 1987.3 2120.9 2250 2471.6 2655.3 2672 2972.6 3340 r 0.2 r 0.12 r 0.02 r 0.1 r 0.3 r 0.01 r 0.01 r 0.2 r 0.4 3540.4 3577.4 3690.7 3774.2 3974.6 3991.3 4241.8 4250 4291.9 r 0.04 r 0.02 r 0.01 r 0.15 r 0.07 r 0.24 r 0.07 r 0.03 r 0.06 4322 4525.7 4655 4698.6 4766 4798.4 4826.3 4856 4881.7 r 0.04 r 0.2 r 0.1 r 0.02 r 0.2 r 0.03 r 0.01 r 0.04 r 0.04 4899.4 4901 4943.2 5003.5 5019.4 5122.3 5146.6 5233 5256.9 r 0.01 r 0.04 r 0.1 r 0.1 r 0.15 r 0.07 r 0.16 r 0.1 r 0.3 5298.6 5308 5310.6 5319.5 5344 5367.7 5401.9 5423 5436.7 r 0.1 r 0.06 r 0.02 r 0.5 r 0.15 r 0.51 r 0.55 r 0.01 r 0.01 252 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS The eigenfrequencies ZSn /2 are determined from the stationary condition on 3 (9.3.2). Table 9.4.2 shows the computed frequencies and the errors obtained by the eigenvalue problem. Resonant vibration modes are excited by applying an external electric field 0 EEE13 exp(i Z0 t ) on both sides of the plate with Z Zn . The undetermined system parameters P are computed by using a genetic algorithm (Chiroiu C. et al. 2000). The number of the measured eigenfrequencies is N 63 Table 9.4.2). The genetic parameters are: number of populations = 45, ratio of reproduction = 1, number of multi-point crossovers = 1, probability of mutation = 0.25 and maximum number of generations = 550. The genetic algorithm exhibits very good convergence and accuracy. For example, for H 0 the maximum values of Z after 344 iteration is 4 6 Zmax 0.51u 10 and G u100.52 in the case of normal modes ZS/2 =334 kHz, and 4 6 Zmax 0.33u 10 and G u100.34 after 275 iteration in the case of ZS/2 =501 kHz. The minimum value for Zmax was found to be Z/4S and the maximum value for maxG 0.11 u 105 forH 0 . Number of iterations varies from 155 to 500. In all computations, the measured eigenfrequencies are computable from the direct problem. A measurement noise has been artificially introduced by multiplication of the data values by 1 r ,r being random numbers uniformly distributed in [,H H] , with H 1012 , 10 , 103 . Results for the case of normal modes ZS/2 =334 kHz are displayed in Table 9.4.3. Table 9.4.3 Maximum value of Z and number of iterations for the normal case ZS/2 =334 kHz. H 0 H 101 H 102 H 103 4 2 3 4 Zmax 0.33u 10 0.256u 10 0.193u 10 0.673u 10 G 0.52u 106 0.119u 102 0.452u 104 0.92u 106 number of 344 386 277 289 iterations Table 9.4.3 shows that Zmax varies linearly with H and G quadratically with H . Other cases have shown also a linear variation of Zmax and a quadratic variation of G with respect to measurement noise. In Figure 9.4.1 the admittance curve ( k / UZ vs. ZS/2 ) in the linear regime 0 ( E # 0.1V ) marks by peaks the frequencies Z Zn of the modes. The agreement between the eigenfrequencies given by this curve and by the eigenvalue problem results (Table 9.4.2) is noted to be excellent. On comparison of the results given by the admittance curve (Figure 9.4.1) with the results obtained from the similar experimental curve derived by Alippi and Crăciun, the deviation between them is found to be 5–15% for low natural frequencies, and less than 4% for high natural frequencies. If E 0 is 0 increased above a threshold value Eth = 5.27 V the Z/2 subharmonic generation is observed. Note that Alippi and Crăciun obtain in the Cantor-like sample typical values of the lowest threshold voltages of 3–5 V. The amplitude of waves is calculated at the surface of the plate as a function of E 0 . INTERMODAL INTERACTION OF WAVES 253 Figures 9.4.2–9.4.4 show the displacements of the normal modes ZS/2 =334 kHz, 501 kHz, 835 kHz and respectively of the subharmonic modes ZS/4 =167 kHz, 250.5 kHz, 417.5 kHz. Two kinds of vibration regimes are found: a localized mode (fracton) regime represented in Figure 9.4.5 for ZS/2 =1169 kHz, 2672 kHz and 3340 kHz and an extended vibration (phonon) regime represented in Figure 9.4.6 for ZS/2 = 4175 kHz and 4250 kHz. A sketch of the plate geometry is given on the abscissa (dashed, piezoelectric ceramic and white, epoxy resin). The fracton vibrations are mostly localized on a few elements, while the phonon vibrations essentially extend to the whole plate. In the case of a periodical plate the dispersion prevents good frequency matching between the fundamental and appropriate subharmonic modes. For the homogeneous plate the mismatch ZZn /2 is due to the symmetry of fundamental modes with respect to x . Only symmetric odd n can induce a subharmonic, but Z/2 never coincides with a plate vibration mode. For a Cantor plate, we have obtained qualitatively the same result as Crăciun et al.: given a normal mode Zn , for excitation at Z Zn , the value of the expected threshold Eth , i. e. the ability of generating the Z/2 subharmonic, is determined by the existence of a normal mode with: (i) small frequency mismatch Zn Z/2, and, (ii) large spatial overlap between the fundamental and subharmonic displacement field. Concerning the values of amplitudes, the results obtained by us verify experimental predictions of resonant amplitudes with accuracy better than six percent for similar fundamental modes. Figure 9.4.1 The admittance–frequency curve for the Cantor plate. Figure 9.4.2 The amplitudes of the surface displacement of the normal mode ZS / 2 = 334 kHz and of the subharmonic mode ZS / 4 = 167 kHz. 254 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS Figure 9.4.3 The amplitudes of the surface displacement of the normal mode ZS / 2 = 501 kHz and of the subharmonic mode ZS / 4 = 250.5 kHz. Figure 9.4.4 The amplitudes of the surface displacement of the normal mode ZS / 2 = 835 kHz and of the subharmonic mode ZS / 4 = 417.5 kHz. INTERMODAL INTERACTION OF WAVES 255 Figure 9.4.5 The normal amplitudes for three localized vibration modes (ZS / 2 =1169 kHz, ZS/2 =2672 kHz and ZS/2 =3340 kHz). Figure 9.4.6 The normal amplitudes for two extended vibration modes (ZS / 2 =4175 kHz and ZS/2 =4250 kHz. 9.5 Internal solitary waves in a stratified fluid Consider an incompressible fluid stratified in density between two horizontal boundaries spaced at distance h apart. The wave motion takes place in the (,x y ) plane of a Cartesian system of coordinates with the origin in the lower boundary and y vertically upward. We review the Yih results in the case of an exponential stratification. The Euler motion equations are given by Du ()UU p , (9.5.1) Dt x 256 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS Dv ()UU pg ( UU) , (9.5.2) Dt y whereu and v are velocity components in the direction of positive x andyp , is the pressure, U(y) is the density in the undisturbed fluid, U the density perturbation and g the gravitational acceleration, and D www uv . (9.5.3) Dttwww x y The incompressibility equation D ()UU 0 , (9.5.4) Dt allows the continuity equation to be written as uvxy 0 . (9.5.5) Therefore, the components of velocity are expressed with respect to the stream function u \y , v \x . (9.5.6) From (9.5.1) and (9.5.2) we have D(\\ + ) Du Dv ()UU xx yy () UU pg p . (9.5.7) DDDttyxt x The equation (9.5.4) becomes DU U\ . (9.5.8) Dt yx Let us introduce now the dimensionless quantities x y tgD xc , yc , tc , h h h \ U U \c , Uc , Uc , (9.5.9) hghD U0 U0 and suppose that the density stratification is defined by U U0 exp( Dy ) . (9.5.10) Dropping the prime, the dimensionless U can be taken as U Dexp( Dy ) T. (9.5.11) The equations (9.5.7) and (9.5.8) are written in terms of dimensionless terms D(\\ + ) Du Dv (1DT ) xx yy () DDT DT T , (9.5.12) DDttyy xDt x INTERMODAL INTERACTION OF WAVES 257 Ttxyuv TT \x . (9.5.13) The solutions for the mn -th and -th modes of the equations (9.5.12) and (9.5.13) are, to the lowest order 1 D \ ( 1)m axtmyy sech2 [ J ( )]{sin S sinm Sy }, T m S\, (9.5.14) mmmmS 2 m m 1 D \ ( 1)n axtnyyn sech2 [ J ( )]{sin S sin Sy } , T nS\, (9.5.15) nnnnS 2 n n maSD maSD2 J2 m , for odd m , J2 m , for even m , (9.5.16) m 9 m 36 2 and similarly for Jn . Consider the simple case when D 0 . In this case, let us assume ˆ \ \mn \ I, T Tmn T T. (9.5.17) Neglecting the higher order terms, the linearised forms of (9.5.12) and (9.5.13) are ˆ 222 ITyyt x ()nm S W, ˆ ITxt ()nmW S , (9.5.18) WP \ny \ mx \ my \ nx sin(n m ) Sy Q sin(nmy ) S , 11 Paanxtxt ( 1)mn S { J sech22 [ J ( )] sech [ J ( )] u mn m nnmSS m 111 uJtanh[ (x tm )] J sech22 [ J ( x t )]sech [ J ( x t )]u (9.5.19) mnmnmmSSnS 1 uJtanh[ (xt )] }, n nS 11 Qaanxtx ( 1)mn SJ { sech22 [ J ( )] sech [ J ( t )]u mn m nnmSS m 111 uJtanh[ (x tm )] J sech22 [ J ( x t )] sech [ J ( x t )]u mnmnmmSSnS 1 uJtanh[ (xt )]}. n nS From (9.5.18) and (9.5.19) we see that we can take I fnmsin( nmy ) S fnm sin( nmy ) S , ˆ T gnmsin(nmyg ) S nm sin( nmy ) S . (9.5.20) 258 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS We treat only the (nm )) mode, since the (nm mode is similar. The terms containing sin(nm )Sy in (9.5.18) are 22 2 2 2 ()nmS f()nmt g () nmx S() n m P, fg()nmx () nmt () nmSP . (9.5.21) By eliminating gnm from (9.5.21), we have 22 2 2 2 ()nmS f()nmtt f () nmxx SS( n m ) Pt () nmPx, (9.5.22a) and similarly, by eliminating fnm from these equations 22 33 2 2 2 ()nmS g()nmtt g () nmxx () nmSS Pt ( n m ) Px. (9.5.22b) The hyperbolic equations (9.5.22) can be solved by integration along the characteristics. We solve only (9.5.22a), the other equation being solved in a similar way. With new variables 1 1 [ x t , K x t , (9.5.23) ()nmS ()nm S the equation (9.5.22a) yields 4[22fmP()nm[K S[KnP ]. (9.5.24) By integrating from t f , we obtain K[ 2dfmPn S KSPd [, (9.5.25) nm ³³ f f or 2 mn 2 §·nnm() nnm () 4(1)()faannmIImnm m n S¨¸ Jm12 Jm H , (9.5.26) ©¹22nm nm where K 111 Ixtxtx sech22 [JJJ ( )] sech [ ( )]tanh[ (t )]dK , 1 ³ nmm f nmSSmS [ 111 Ixtxtx sech22 [J ( )] sech [ JJ ( )]tanh[ (t )]d[ , (9.5.27) 2 ³ nmm f nmSSmS 1 1 H sech22 [J (xt )] sech [ J ( xt )] . mnmnSS The computations have shown that integrals I1 and I2 are convergent. In particular, fornm 2 , we obtain INTERMODAL INTERACTION OF WAVES 259 111 I 8mxtxtx22 sech {JJJ ( )]tanh[ ( )]tanh[ (t )]} . (9.5.28) 1 mmnmmSSnS The interaction of waves can be discussed looking at (9.5.26). The first part in (9.5.26) may be eventually independent of K and represents a wave propagating in the positive direction of x . The second term may be independent of [ and represents a wave propagating in the opposite direction. The last term in (9.5.26) is zero as t increases, since it is the product of two squares of the sech function with different 1 arguments. These waves propagate with velocities r . ()nmS It is convenient to discuss the interaction of waves for the particular case of nm 2 . These waves may propagate in the same direction or in opposite directions. The result of interactions of waves are two pairs of solitary waves, each pair consisting of two opposite directions propagating solitary waves of the same mode. The modes of the two pairs are different from each other, and are different from the m-th and n-th modes of the waves. The original waves propagate after interaction without changing their identities, but only the m -wave suffers a shift of phase. In the same manner, the analysis may be applied for the general case of Dz0 . The interaction of more than two waves can be dealt with pair by pair (Hirota and Satsuma). 9.6 The motion of a micropolar fluid in inclined open channels Consider a two-dimensional flow of a micropolar, isotropic, incompressible, viscous fluid in a wide channel over a rigid bottom (Chiroiu et al. 2003b). We have chosen a wide channel to be sure that the motion will be two-dimensional only. The x-axis is horizontal and the bottom is given by h(x). The channel bed is linear and is inclined at an angle T!0 below the horizontal, that is ymx , with m tanT! 0 (Figure 9.6.1). The vertical distance of the surface above the x-axis is denoted by K (x). The fluid domain : is a two-dimensional strip :f :,()x f 0 Kyx, bounded by a free surface *: y K , and a lower rigid bottom S : hx() mx . In the shallow flow the vertical dimensions are small compared to the horizontal dimensions. The motion equations of a micropolar, viscous fluid are given by Eringen (1966, 1970): UUvv grad vX grad SPD( )curl curl v (9.6.1) PO(2 )grad div vw D 2 curl , UUJwJvwY grad JH( )curl curl w (9.6.2) J](2 )grad div ww DD 4 2 curl v , where X (,XX12 ) is the exterior body force, YYY (,12 ) is the exterior body couple, S is the thermodynamic pressure, UJ is the inertia tensor density, v (,vv12 ) is the w velocity vector vu , uu (,u ) the displacement vector, I (, M M ) the wt 12 1 2 260 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS w microrotation vector, www (, ) the microrotation velocity w I,U the fluid 12 wt density. The superposed dot indicates the partial differentiation with respect to time w a a . In (9.6.1) and (9.6.2) O and P are the classical viscosities coefficients of the wt Navier–Stokes theory. The constants D,,] J and H are the micropolar coefficients of viscosity. The elastic coefficients must fulfil the condition (Eringen 1970) POtDttP0, 2 30,0, (9.6.3) Jt0, 2 J30,0.]t Ht Equations (9.6.2) and (9.6.3) are six equations with unknown vector fields the velocity v and microrotation w . These equations must be supplemented by the equation of continuity U div( Uv ) 0 . (9.6.4) Here we consider an incompressible fluid with U constant. From (2.5) we obtain div v 0. (9.6.5) In this case, the thermodynamic pressure S must be replaced by an unknown pressurep to be determined through the solution of each problem. The constitutive relations are Vij (Opvkk,,,) Gij ()()2P Dvvwj i PDi j DH kij k , (9.6.6) Pij ]wwkk, GJij ()Hji, () JH wij, , (9.6.7) whereVij is the stress tensor and Pij is the couple stress tensor. The field of equations (9.6.1), (9.6.2) and (9.6.5) are subject to certain boundary and initial conditions: – Traction conditions on * Vklntk l , Pklnk P l , (9.6.8) wheretl are the surface traction and Pl the surface couple acting on * . Here nk ,k 1, 2(,) are the unit vectors of the coordinate system x y . – The condition for a particle at the surface * to remain at the surface * K v1, Kx v2 . (9.6.9) – Velocity conditions of adherence of the fluid to S 0 0 vxkkk(,tw) vxtw,(,), k (9.6.10) 00 where x S and vkk, w the given values for velocity and microrotation velocity. – The initial conditions at t 0 vvw 00,, w K K0 . (9.6.11) INTERMODAL INTERACTION OF WAVES 261 rvv22UU|| rvv || We consider that XX 11 11,, UgYY 0 in 12R K 12 (9.6.1) and (9.6.2). Therefore, equations (9.6.1), (9.6.2) and (9.6.5) are then rvv2U || UUvvvvvp U PD'() v 11, (9.6.12) 111,21,,xyx 1 K Uvvvvvp212,22,,UUxyy PD'U() v2g , (9.6.13) UUUJw111,21, Jv wxy Jv w (2J] ) w 1,xx (9.6.14) J]H()()4,www2,xy JH1,yy D 1 UUUJw 212,22, Jv wxy Jv w (2J] )w2,yy (9.6.15) J]H()()4,www1, xy JH2, xx D 2 vv2,xy 1, 0 , (9.6.16) ww2,xy 1, 0 , (9.6.17) vv1,xy 2, 0 . (9.6.18) The comma represents the differentiation with respect to the shown variable. In (9.6.13), g is the constant gravitational acceleration. In the right side of (9.6.12) the rv2 ||v term 1 1 represents the resisting body force always acting opposite to that of the R flow, where r 2 is a constant depending upon the roughness of the channel walls and R is the hydraulic radius. Figure 9.6.1 Geometry of the flow (Dressler). According to this formula, the turbulent fluctuations exert on the main flow a rv22 resistive body force at every point of magnitude 1 . Since the most flows in practice R are highly turbulent we take account of the resistive force due to the momentum transport of the secondary flow exerted on the average flow at each point. The resistance effects due to the dynamic viscosity of the water are neglected. The above expression of the resisting force was given by Chezy (Dressler). The hydraulic radius is 262 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS defined as the ratio of the area of a cross-section of the water normal to the channel to its wetted perimeter, that is that part of the perimeter excluding the free surface of the water. The Chezy formula expresses the fact that the resistance will be greater in shallow regions where all of the water is closer to the rough boundary. The Chezy formula is valid only for uniform flows, and although it is used for non-uniform flows when the flow varies slowly with respect to x, y and t. In our case R = K. In (9.6.12)– (9.6.18) the unknown functions are vwiii,, 1,2 ,andp K. We attach to (9.6.12)–(9.6.18) the conditions (9.6.8)–(9.6.11) VV11n 1 21nt 2 1 , VV12nnt 1 22 2 2 , on * , (9.6.19) P11nn 1P 21 2 P 1 , PPP12nn 1 22 2 2 , on * , (9.6.20) K vv1, Kx 2, on * , (9.6.21) 00 0 0 vv11 ,, v 2 vww211 , w2 w2 onS , (9.6.22) vv1102 ,, vv 20 w11022 www, 0 ,K K0 at t 0 . (9.6.23) In (9.6.19)–(9.6.20) the stress and couple stress components are given by (9.6.6)– (9.6.7). Let the constant H be a typical vertical dimension of the resulting flow, and L a H 2 typical horizontal dimension. Let G be a small positive parameter, which we will L2 use for a perturbation procedure. New dimensionless variables are defined by xy gH D ,,E W t , LH L K p v1 h ,,P V1 , d , HgHU gH H Hv21 w Hw 2 VWW212 ,, , (9.6.24) LgH gH//L gH PDrH2 P , D , r 2 , UULgH LgH L ]JH ] ,,J H . UUULJ gH LJ gH LJ gH In terms of the dimensionless variables (9.6.12)–(9.6.23) become 22 GYV[()]()1,WD VV 1 1, PD V1, DDDE P , VV 2 1, Y PD V1, EE Y rV1 0 , (9.6.25) G[()1VVV2,WD 1 2, PDPD V2, DDE P , ]() VV2 2, E V2, EE 0 , (9.6.26) INTERMODAL INTERACTION OF WAVES 263 G[(WVW J2)4]] WD WVW 1,WD 1 1, 1, DD1 2 1,E (9.6.27) JH()WW1,EE J]H ( )2,DE 0, G[()4]WVW2,WD 1 2, JHD W2, DD WVW2 2 2,E (9.6.28) (2 J] )WW2,EE ( J]H )1,DE 0, VV2,DE 1, 0 , (9.6.29) WW2,DE 1, 0 , (9.6.30) GVV1,DE 2, 0 , (9.6.31) VV11nn 1 21 2 t 1 , VV12nn 1 22 2 t 2 , at E Y , (9.6.32) PP11nn 1 21 2 P 1 , PP12nn 1 22 2 P 2 , at E Y , (9.6.33) G()YVYV,1,WD 2 , at E Y , (9.6.34) 00 0 0 vv11 ,, vv 2 2 ww 1 1 , ww 2 2, at E d , (9.6.35) GVm12/ H V , at E d , (9.6.36) vv1 10,, vv 2 20 ww 1 10 , ww 2 20 ,K K0 ,0 at t . (9.6.37) We assume that the unknowns can be expressed as power series in terms of G f f ()kk ()kk VVii ¦ (,,)DEWG , i 1, 2 , WWii ¦ (,,)DEWG, i 1, 2 , k 0 k 0 f f PP ¦ ()kk(,,)DEWG , YY ¦ ()k (,)DWGk . (9.6.38) k 0 k 0 We introduce (9.6.38) into (9.6.25)–(9.6.37) and the resulting coefficients of like powers of G are equated (Nayfeh, Kamel). We obtain 1. From the coefficients for G0 (0) (0) (0) (0) (0) (0) 2 (0)2 LYVV1 {PD2 1,EE() YVrV1, E1 0 , (0) (0) (0) (0) (0) LVV2{PD 2 2,EE() YV2, E 0 , (0) (0) (0) (0) (0) LVW3{ 2 1,EE () JH W1, ED ( J] H ) W2, E 0 , (0) (0) (0) (0) (0) LVW4{J 2 2,EE(2] ) W2, EJ (] H ) W1,DE 0 , 264 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS (0) (0) (0) (0) (0) (0) LVV52,1,{DE 0 , LWW62,1,{DE 0 , (9.6.39) (0) (0) (0) (0) (0) LV72,{E 0 , LV82{E Y 0 , (0) (0) (0) (0) (0) (0) (0) LPY9 {E 0 , LV10{E 2 d 0 , LW11{E 1 d 0 . 2. From the coefficients for G1 (1) (0) (0) (0) (0) (0) (0) (0) (0) (0) LYVVVY1 {1,WD 1 1, PD() YVPY1,DDE , (1) (0) (0) (0) (1) (0) (0) (1) PDVV2 1,EE Y VVY2 1, () YV1, EE (0) (0) (1) (1) (0) 2 (0) (1) PDVVY21,EE() YV1,E 2 rVV 11 0, (1) (0) (0) (0) (0) (0) LV2{ 2,WD VV 1 2, PD() V2, DDE P , 1 (1)(0) (0)(1) (1) VV22,2EE VV 2, PD() V2, EE 0, (1) (0) (0) (0) (0) (0) (1) (0) LW3{ 1,WD VW 1 1, J(2] ) W1, DDD 4 W1 VW 2 1,E (0) (1) (1) (1) VW21,EE () JH W1,ED ( J]H )W2,E 0, (1) (0) (0) (0) (0) (0) (0) (1) L4{ W 2,WD VW 1 2, JHD() W2, DD 4 W2 VW 2 2,E (1) (0) (1) (1) VW22,EE (2 J] ) W2,ED ( J]H )W1,E 0, (1) (1) (1) (1) (1) (1) (1) (0) (1) LVV52,1,{DE 0 , LWW62,1,{DE 0 , LV71,2,{ DEV 0 , (9.6.40) (1) (0) (0) (0) (1) (1) (1) (1) (1) LY8,{WD VYVY 1,2 0 , LPY9 {E 0 , (1) (0) (1) (1) (1) (1) (1) LVmHV10{ 1 /02 , LW11{E 1 d 0 , LW12{E 2 d 0 . 3. From the coefficients for G2 (2) (0) (1) (0) (1) (0) (0) (1) (1) (0) (0) LYVVYVVYVVY11,1,11,11,{WW DD (0) (1) (0) (1) (0) (1) (0) (1) (1) (0) (0) (2) VVY1 1,DE VVY2 1, VVY2 1, EE VVY2 1, (1)(0) (0)(1) () PDYV1,DD () PD YV1,DD (1) (0) (0) (1) (0) (2) (0) (2) (0) PY,,21DD PY V V, E Y PD() Y V1,EE (1)(1) (0)(2) 2 (1)2 (0)(2) () PDYV1,EE () PD YV1,EE rV (1 2 VV1 1 ) 0, (2) (1) (0)(1) (1)(0) (0)(2) L2{ V 2,WDDE VV 1 2, VV 1 2, VV 2 2, (2)(0) (1)(1) (1) (0)(2) VV2 2,EE VV 2 2, PD() V2, DDE PD () YV1, E 0, INTERMODAL INTERACTION OF WAVES 265 (2) (1) (0) (0) (0) (1) (2) (0) L3{ W 1,WDDE VW 1 1, VW 1 1, VW 2 1, (1)(1)(0)(2) (1) (1) (2) VW2 1,EE VW 2 1, O(2] ) W 1, DDD 4 W 1 JH ( ) W 1,EE (2) J]H()0,W2,EE (2) (1)(1)(0)(0)(1)(0)(2) L4{ W 2,WD VW 1 2, VW 1 2, DE VW 2 2, (2)(0)(1)(1) (2) (1) (1) JVW2 2,EE VW 2 2, (2] ) W2, EED 4 W2 JH ( ) W2,DD (2) J]H()0,W1,DE (2) (2) (2) LVV52,1,{DE 0 , (9.6.41) (2) (2) (2) LWW62,1,{DE 0 , (2) (1) (2) LVV71,2,{DE 0 , (2) (1) (0)(1) (1)(0) (2) (2) LYVYVYVY8,{WDD 1,1,2 0 , (2) (2) (2) (2) (1) (2) LPY9 {E 0 , LVmHV10{ 1 /02 , (2) (2) (2) (2) LW11{E 1 d 0 , LW12{E 2 d 0 . 9.7 Cnoidal solutions Consider that wk Mf()kk log() (D ,EW , ) , kN 1,2,..., , i 1,2,...,6 , (9.7.1) iiwDk where M (,VVWW12 , 1 , 2 , PY , )and (1) fii(,,)DEW 1 exp T1 , (2) fii(DEW , , ) 1TTTT exp12 expi exp( 1ii2 ) , (9.7.2) ………. NN N ()N fi(D , E , W ) 1 ¦¦ exp Tji exp( Tjili T ) ¦ exp( Tjiliri T T ) .... , jj 11zl j zlzr 1 with Tki ab ki D ki EZ ki W ki , kN 1,2,..., , i 1,2,...,6 , (9.7.3) andabki, ki the dimensionless wave numbers, Zki the dimensionless frequencies and ki the dimensionless phases. We find that asymptotically the solutions become 266 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS ()k 2 MAiikkikikik sech ( abDEZ' t2i ) , (9.7.4) with kN 1,2,..., ,i 1,2,...,6 , as t orf . ()k The functions M i are periodic with the period 2'ki . These solutions represent a linear superposition of soliton waves, which is a row of solitons, spaced 2'ki apart. The 23u N parameters in this formulation abki, ki , Zki and ki ,,kN 1,2,..., i 1,2,...,6 are computable by substituting (9.7.1) in (9.6.39)–(9.6.41). The parameters are defined by pab {,,ki kiZ ki ,} ki , kN 1, 2, ... , i 1,2,...,6 . (9.7.5) The wave numbers, frequencies and constant phases are also vectors aaaaaki (11 , 12 , 13 ,.....N 6 ), bbbbb ( , , ,.... ), ki 11 12 13N 5 (9.7.6) Zki (ZZ11 , 12 , Z 13 ,.... ZN 6 ), ki (11 , 12 , 13 ,....N 6 ). The resulting system is a system of 36 equations to determine a number of 23u N unknowns. Details of the genetic algorithm can be found in Goldberg. It is assumed the parameters p are discretized into discrete values with the step width 'pab {,, 'ki 'ki 'Zki , ' ki } . The set of parameters for arbitrary values pa {,,ki,, mb kin Z ki ,, q ,} ki s can be expressed as 6N numbers NmNQSnQSqSikmnqs (1)ik ik ik (1)(1)ik ik ik s , where M ki ,NQki, ki and Ski denote the total number of discretized values for each parameter p . These numbers represent an individual in a population and for the discretized parameters indicate a specific solution. An individual is expressed as a row of the integer number with Ngen = 6N genes. A fitness value is evaluated for each individual and in the total population only individuals with a higher fitness remain at the next generation. The alternation of generations is stopped when convergence is detected. If no convergence the iteration process continues until the specified maximum number of generations is reached. To compute the fitness F we write (9.6.39)–(9.6.41) in the form ()mm () Lkk S , m 0,1,2 , k 1,2,...,12 . (9.7.7) ()m ()m and note the square sum of differences LkkS by 212 (L()jj ()) 2. (9.7.8) ¦¦ kk S jk 01 We define fitness as follows INTERMODAL INTERACTION OF WAVES 267 F 0 , (9.7.9) with 212 ()()j 2. (9.7.10) 0 ¦¦ Sk jk 01 As the convergence criterion of iterative computations we use (9.4.14) 1 Z log 0 o max . 2 10 Numerical simulation is carried out for O =1.055u 10 3 kg/ms, and P =1.205u 10 3 kg/ms. The micropolar coefficients of viscosity have values D = ] =H =1.035u 103 mkg/s (Gauthier). We consider m tan T, with m [0.2, 0.8]. The value m = 0.8 represents an upper limit on the slopes for which the shallow fluid theory furnish a good approximation. The number r 2 must satisfy the condition (Dressler) 40.7r 2 d m , (9.7.11) which is important for existence of waves. If the resistance is too large, the waves cannot form. This condition is obtained numerically. We take r 2 [0.035, 0.14]. The value r 2 = 0.14 was chosen as the greatest value for the resistance since it satisfies the condition (9.7.11) for m = 0.8. The intervals for the model parameters are evaluated from the condition that the total mass of fluid per wavelength is constant and the same in all approximations. In order to illustrate the results three cases are considered ( N 4 ): – Case 1. T 45D (m = 1), r 2 = 0.17, – Case 2. T 31D (m = 0.6), r 2 = 0.1, (9.7.12) – Case 3. T 22D (m = 0.4), r 2 = 0.06. In all cases we have assumed that the number of populations is 25, ratio of reproduction is 1, number of multi-point crossovers is 1, probability of mutation is 0.2 and maximum number of generations is 250. The linear summation of the solution Y (,)DW for Wof is given in Figures 9.7.1– 9.7.3 (Wof means in the numerical simulation the time interval after which the solutions have a permanent profile in time). In all cases the fluid velocity is greater in the region of the crests than in the shallower regions, but nowhere will the fluid velocity be as great as the wave speed. For example, in Case 1 the average fluid velocity is about 3.05 m/s while the wave velocity is about 4.1 m/s. 268 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS Figure 9.7.1 Profile of the wave Y (,)DW in Case 1. Figure 9.7.2 Profile of the wave Y(,)DW in Case 2 INTERMODAL INTERACTION OF WAVES 269 Figure 9.7.3 Profile of the wave Y(,)DW in Case 3 From numerical simulations we conclude that the remaining solutions have a similar evolution with respect to D : they increase and decrease in the same manner as Y . The microrotation components and the vertical component of the fluid velocity are greater in the crest regions than in the shallower regions. The model parameters were obtained after 149 iterations in Case 1, 167 iterations in Case 2 and 187, in the last case. In conclusion, the solutions describe the phenomenon called roll-waves for fluid flow along a wide inclined channel. This phenomenon appears in hydraulic applications, such as run-off channels and open aquaducts. When a liquid flows turbulently downwards on an inclined open channel, the wave profile represented as sums of solitons moves downstream as a progressing wave at a constant speed and without distortion, and such that the velocities of the fluid particles are everywhere less than the wave velocity. 9.8 The effect of surface tension on the solitary waves In this section we discuss the effect of surface tension on the solitary waves. This effect may be important when the depth is small enough and the surface tension is present. This effect was analyzed by Shinbrot in 1981, and the present discussion is principally based on his results. Let h be the depth of the fluid at infinity, Ug the weight density of the fluid, and T the surface tension. The fluid moves over a flat bottom being acted by the gravity force. The system of coordinates is moving horizontally with the phase speed of the flow, the x - axis is lying at the bottom and the y- axis is vertical. The free surface is described by the function yHx (). The velocity of the fluid is denoted by Sxy(,), (U V ). The uniform flow of the fluid is described by the equation of irrotationality and incompressibility of the fluid (Camenschi) 8 UVyx , UVxy 0 , for 0 yHx(), (9. .1) by the streamlines at the top and bottom 270 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS V 0 , for y 0 , (9.8.2) 8 VUH x , for yHx (), (9. .3) the condition that the pressure at the free surface is proportional to the curvature H T xx 1 22 8 gH 23/2 ()conUV st. , for yHx (), (9. .4) U(1H x ) 2 and the condition of solitary waves (Nettel) Hx()o const., Ux(,y )o const., Vx(,y )o 0, as ||x of. (9.8.5) Shinbrot maps the domain of the fluid into the strip {(XY , ) : f X f, f Y f}, Hx y X , Y , (9.8.6) h H ()x whereh lim||x of Hx ( ) , and H is a small parameter, measuring the ratio of the amplitude of the wave to the depth at infinity. Suppose H ()x ,Uxy(, ) and Vx(,y ) have the form H()xh [1HK2 ( X )], (9.8.7) 2 8 Uxy(, ) U0 [1H uXY ( , )], (9. .8) 3 8 Vxy(, ) U0H vXY ( , ). (9. .9) Here, the function K is continuous as H tends to zero, and suppose we have the normalized condition 8 supX |K (X ) | 1 , (9. .10) 8 UU0|| limx of (xy , ) . (9. .11) The equations (9.8.1)–(9.8.5) become 24 8 uvYX H H() K vYv XX K Y , for 0 Y 1 , (9. .12) 2 8 uvX YX HK() YuY K uX , for 0 Y 1 , (9. .13) v 0 , for Y 0 , (9.8.14) 2 8 vu KXX H K , for Y 1 , (9. .15) 2 24 HWKXX HH22 8 K623/2 Fu() u v 01, for Y , (9. .16) (1H KX ) 22 INTERMODAL INTERACTION OF WAVES 271 K()X ,,uXY(,)vXY(,)o 0 , as||X of . (9.8.17) In (9.8.16),W is the inverse Bond number given by T W , (9.8.18) Ugh2 and F is the Froude number which is not known since U0 is not known. U 2 F 0 . (9.8.19) gh Next, we insert the following expansions in series of powers of H2 (Nayfeh, Camenschi and ùandru) uu (0)H 2 u (1) ..., vv (0)H 2v (1) ..., FF (0)H 2 F (1) ..., K K(0) H 2 K (1) ... , into (9.8.12)–(9.8.17) and equate coefficients of like powers of H . The equations for zero order approximations are (0) (0) (0) 8 uY 0 , uvXY 0 , for 0 Y 1 , (9. .20) v(0) 0 , for Y 0 , (9.8.21) (0) (0) 8 v KX , for Y 1 , (9. .22) K(0)Fu (0) (0) 0 , for Y 1 , (9.8.23) K(0) , u(0),v (0) o 0 , as||X of . (9.8.24) In these equationsK(0) is free, being subjected to (9.8.24). The solutions of the equations (9.8.20)(9.8.23) are given by (0) (0) (0) (0) (0) 8 F 1 , uXY(,) K () X, vXYY(,) KX () X. (9. .25) From (9.8.25), the equations for the next approximation are (1) (0) (1) (1) (0) (0) 8 uYuYX X , uvX YX KK, for 0 Y 1 , (9. .26) v(1) 0 , for Y 0 , (9.8.27) (1) (1) (0) (0) 8 v KX K KX , for Y 1 , (9. .28) 1 2 KWK(1) (1)Fu (1) K (0) (1) K (0) 0 , for Y 1 , (9.8.29) XX 2 K(1) ,,u(1) v(1) o 0 , as||X of. (9.8.30) 272 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS We obtain from (9.8.26)–(9.8.28) Y 2 uXY(1)(,) fX () K (0) () X, (9.8.31) 2 XX Y 3 VXYyfX(1)(,) [ ()KKK (0) () X (0) ()] X (0) () X. (9.8.32) XX6 XXX In (9.8.31), the functionf is free. Substituting (9.8.31) and (9.8.32) into (9.8.29) andtheninto (9.8.28) it results that K(0) satisfies the equation 1 ()W K(0) FX (1) K (0) ()3 K(0) K (0) . (9.8.33) 3 XXX X X IfW 1/3 , the equation (9.8.33) yields to K(0) const. From (9.8.7) we obtain H const. , but this result violates the original hypothesis. For Wz1/3, by integrating twice the equation (9.8.33) we have 1 2 2 ()W K(0) (F (1) K (0) ) K (0) . (9.8.34) 3 X In this case, the solution of (9.8.34) is not finite for all x . The conclusion is that the solitary waves do not exist for Wt1/3. For W 1/3 the solution of (9.8.34) is obtained by using (9.8.24) 4a2 K(0) (1 W 3 )sech2 (aX ) . (9.8.35) 3 By neglecting the terms of the order H44a , we obtain 4 Fa 1(1 H22 W3), (9.8.36) 3 4 HaX Hh [1HW22 a (1 3 )sech2 ] . (9.8.37) 3 h ForW 0 , the solutions (9.8.35) and (9.8.36) reduce to the solutions with no surface tension. The effect of surface tension is to reduce the quantity (1F ) by a factor (1W 3 ) . Also, the amplitude of the soliton is reduced with the same factor. By imposing the condition (9.8.10) to the solution (9.8.35) we obtain 4(13)3a2 W . Introducing this value ofa into (9.8.36) and (9.8.37) we obtain HX 3 F 1H2 , Hh [1H22 sech ] . (9.8.38) 213h W From (9.8.38) we see that for fixed amplitude, the effect of surface tension is to sharpen the crest of the wave by shortening its length by the factor 13W. Chapter 10 ON THE TZITZEICA SURFACES AND SOME RELATED PROBLEMS 10.1 Scope of the chapter The study of affine differential geometry was initiated by Gheorghe Tzitzeica (1873–1939) in 1907 by studying a particular class of hyperbolic surfaces. Tzitzeica proved that the surfaces for which the ratio K / d 4 ( K is the Gaussian curvature andd , the distance from the origin to the tangent plane at an arbitrary point) is constant, are invariants under the group of centroaffine transformations. The Tzitzeica property proves to be invariant under affine transformations, and his surfaces are called Tzitzeica surfaces by Gheorghiu, or affine spheres by Blaschke because they are analogues of spheres in affine differential geometry, or projective spheres by Wilczynski (see Bâlă). Bâlă (1999) applied the symmetry groups theory to study the partial differential equations which arise in Tzitzeica surfaces theory. In this chapter, a survey of the Tzitzeica surfaces and the application of the symmetry group theory to the Tzitzeica equations, are presented (Bâlă). The chapter explains the Shen–Ling method to construct a Weierstrass elliptic function from the solutions of the Van der Pol’s equation. Next, an application in the field of mechanics, closely related to the Tzitzeica equation, is included. In the last section, the results of Rogers and Schief concerning the capability of the Bäcklund transformation to provide an integrable discretization of the characteristic equations associated to the problem of an anisentropic gas, are considered. We refer the readers to the articles by Bâlă (1999), Crăúmăreanu (2002), Musette et al. (2001), Rogers and Schief (1997, 2002), Shen (1967) and Ling (1981). 10.2 Tzitzeica surfaces LetD R 2 be an open set and consider a surface 6 in R3 defined by the position vector ruv(,) ruv(,) xuvi (,) yuv (,) j zuvk (,) , (,)uv D. The vector ruv(,) , which satisfies the condition 274 ON THE TZITZEICA SURFACES AND SOME RELATED PROBLEMS (,rruv , r )z 0, (10.2.1) can be considered the solution of the second-order partial differential equations system that defines a surface leaving a centroaffinity aside rarbrcruu u v , rarbrcruv ccc u v , (10.2.2) rarbrcrvv cc u cc v cc , which is completely integrable, that is ()rruu v () uv u , ()rruv v () vv u , (10.2.3) whereaa,ccc , a ... are the centroaffine invariant functions of uv and . For cc cc 0 , the surface 6 is related to the asymptotic lines. THEOREM 10.2.1 (Tzitzeica) Let 6 be a surface related to the asymptotic lines. K The ration I is a constant if and only if abcc 0 . d 4 Therefore, the Tzitzeica surfaces are defined by the system of equations rarbruu u v , rhruv , (10.2.4) rarbrvv cc u cc v , wherecc h . The integrability conditions (10.2.3) become ah hu , abahv cc , bbv bcc 0 , (10.2.5) bhcc hv , aaacc cc 0 , babhucc cc . We have met these equations in Section 2.4, in the study of the symmetry group of the Tzitzeica equations of the surface. If h satisfies the Liouville–Tzitzeica equation (lnhh )uv , (10.2.6) the Tzitzeica surfaces which are not ruled surfaces are defined by ON THE TZITZEICA SURFACES AND SOME RELATED PROBLEMS 275 h M()u h rr u r , rh r , r v r . (10.2.7) uuhh u v uv vvh v If h satisfies the Tzitzeica equation 1 (lnhh ) , (10.2.8) uv h2 the Tzitzeica surfaces which are not ruled surfaces are defined by h 1 rr u r , uuhh u v rhuv r , 1 h rr v r . (10.2.9) vvhh u v The system (10.2.4) can be written in the form Tuu abTT u v , Tuv hT, (10.2.10) Tvv abcc TT u cc v , with the condition that the three independent solutions x xuv(,),,yyuv (,) zzuv (,) of (10.2.10) and (10.2.5) define a Tzitzeica surface. It is known that every linear combination of x,,yzis a solution of (10.2.10). An equivalent form of (10.2.4) is given by xuu ax u bx v , xuv hx , xvv axcc u bx cc v , yaybyuu u v , yhyuv , (10.2.11) yaybyvv cc u cc v , zazbzuu u v , zhzuv , zazbzvv cc u cc v , 276 ON THE TZITZEICA SURFACES AND SOME RELATED PROBLEMS with the conditions (10.2.1) and (10.2.5). This form is useful for studying the symmetries of the system (10.2.4). 10.3 Symmetry group theory applied to Tzitzeica equations The condition (10.2.1) is written as ()()()yzuv zy u v x xz uv xz vu y xy u v yx uv z f, fuv(,)z 0. (10.3.1) LetDUu (2) be the second-order jet space associated to (10.2.11) and (10.3.1). The independent variables u x1 , vx 2 , and the dependent variables x u1 , yu 2 , zu 3 are considered. Let M uD U be an open set and ] , K, I ,O and \ the functions of uv , , x ,yz and . The infinitesimal generator of the symmetry group G of (10.2.11) and (10.3.1) is wwwww X ] K I O \ , wwwwwuvxy z denoted by GG1 a subgroup of on the variables x ,yz and . For ] 0 , K 0 and I(,x yz ,) a11 x a 12 y a 13 z, O(,x yz ,) a21 x a 22 y a 23 z, \(,x yz ,) a31 x a 32 y() a 11 a 22 z, the infinitesimal generator Y of G1 is YaYaYaYaYaYaYaYaY 11 1 22 2 12 3 13 4 21 5 23 6 31 7 32 8 , (10.3.2) with ww Yx z , 1 wwx z ww Yy z, 2 wwyz w w Yy , Yz , 3 wx 4 wx w w Yx , Yz , 5 wy 6 wy w w Yx , Yy . (10.3.3) 7 wz 8 wz ON THE TZITZEICA SURFACES AND SOME RELATED PROBLEMS 277 The Lie algebra associated to G1 of the full symmetry group G of (10.2.11) and (10.3.1) is generated by Yi , i 1, 2,...,8 , and thus the Lie subgroup G1 is the unimodular subgroup of the group of centroaffine transformations. If we consider the subalgebra described by Y12 and Y , then the function F is invariant and satisfies YF1 () 0 and YF2 () 0. Therefore, F M(,,u v xyz ), and the group invariant solutions are Cu 1 , vC 2 and xyz C3 . We obtain from here the known surface of Tzitzeica C z , C R . (10.3.4) xy THEOREM 10.3.1 (Bâlă) The general vector field of the algebra of the infinitesimal symmetries associated to the subgroup G2 , where G2 is the subgroup of the full symmetry group G of (10.2.11), which acts on the space of the independent variables, is ww Zu ]()K () v, wwuv where the functions ] and K satisfy the equations ]aaauvuuu K ] ] 0 , ]bbbbuvu K K 20] u , ]hhhuv K ()] uv K 0 , ]aaauvcc K cc cc] u 2 a ccK v 0 , (10.3.5) ]bbbuvcc K cc ccK vvv K 0 , and the functions abha,,,cc , bcc satisfy (10.2.5). Consider the cases: Case 1. 6 is a ruled Tzitzeica surface given by (10.2.7). The completely integrable conditions (10.2.5) turn in h M()u h a u , b , acc 0 , bcc v , (10.3.6) h h h with h a solution of the Liouville–Tzitzeica equation (10.2.6). The relations (10.3.5) become ]hhhuv K ()] uv K 0 , k ]3 , hh h h h3 . (10.3.7) M uv u v 1 1 Writing ] , K , where UU ()u and VV ()v , then the first equation U c V c of (10.3.7) is 278 ON THE TZITZEICA SURFACES AND SOME RELATED PROBLEMS hUVUV ccP( ) . (10.3.8) Substituting (10.3.8) in the last equation (10.3.7) we have the equation PPcc P c 2 P3 . (10.3.9) The general solution of (10.3.9) is 2 ,0,k ° ()2 ° UVC 2 ° l 2 P()UV ® 2 ,,k l (10.3.10) ° 2cos [0.5lU ( V ) C ] ° 2 l 2 ° 2 ,,kll !0. ¯°2sinh [0.5lU ( V ) C ] From (10.3.8) and the change of the functions UF ()U , VG ()V , we have UUC , VV , for k 0 , l l UU tanh (C ) , VV tanh , for kl 2 , (10.3.11) 2 2 l l UU cotan (C ) , VV tan , for kl 2 . 2 2 Therefore, the general solution of the Liouville–Tzitzeica equation is 2UVcc huv(,) . (10.3.12) ()UV 2 This solution is expressed in terms of solitons for kl 2 , and ll UUC c sech2 ( ) , 22 ll VVc sech2 . 22 Case 2. 6 is a Tzitzeica surface which is not a ruled surface, that is (10.2.9). Then (10.2.5) become h 1 h a u , ba cc , bcc v , (10.3.13) h h h whereh is a solution of the Tzitzeica equation (10.2.8). Substitution of (10.3.13) into (10.3.5) yield ]u 0 , Kv 0 , ]hhhuvK ()] uvK 0 . (10.3.14) ON THE TZITZEICA SURFACES AND SOME RELATED PROBLEMS 279 It results ] C1 , K C2 and hCvCu P( 12 ) , ww ZC C . (10.3.15) 12wwuv Substitution of the first equation of (10.3.15) into the Tzitzeica equation (10.2.8), gives the equation 23 PPPCC12()cc c P1. (10.3.16) For CC12 0 , we have P 11, h . This is a case of Tzitzeica solution (Tzitzeica 1 1924). For CC12z 0 , and denoting k , the equation (10.3.16) becomes CC12 PPPcc c 23 k(1 P) . (10.3.17) Fork 1 , the equation (10.3.17) is a Weierstrass equation Pc232 21,PPCC R . (10.3.18) l Using the change of function P g , (10.3.18) turns in 2 ggCgc23 2 4 . (10.3.19) If O is a real solution of the right side polynomials of (10.3.19), then Oz0 is not a triple solution. Therefore, (10.3.19) yields 44 gggc22 ()(O g ) . (10.3.20) 2 OO ForO 1 , then C 3, and (10.3.19) becomes gggc2 (1)(2) 2 . (10.3.21) 1 Writing g 2 , we have w2 1 wwc22 (3 1) , (10.3.22) 4 with the solution 1()3uv wu() v sinh[CC11 ], R . (10.3.23) 3 2 From here the other solution of the Tzitzeica equation is obtained, by writing 1 h 1, and C 0 2w2 1 280 ON THE TZITZEICA SURFACES AND SOME RELATED PROBLEMS 3 hu() v 1. (10.3.24) 2sinh2 [(uv ) 3/2] ForOz1 , the polynomial of (10.3.19) admits three distinct real solutions for O! (C 31 ), and one real and two complex for O 1 (C !3 ). In this case the integral dg J , ³ 44 ()(ggO2 g ) O2 O is reduced to an elliptical integral of first-order dM J . ³ 2 1sinMk Thus, for C z3 , the solutions of the Tzitzeica equations with the form huv P( ) are given in terms of elliptical functions. PROPOSITION 10.3.1 (Bâlă) The solution (10.3.24) is a revolution Tzitzeica surface. Moreover it is an associated ruled Tzitzeica surface. The proof is given by Bâlă. Tzitzeica studied the revolution surfaces defined by (10.2.10). From the condition that h must verify hhu v , he obtains huv P( ) , and the equation (10.3.17) for k 1 PPcc P c 23 P 1. Tzitzeica found the solution of (10.2.10) by using PPc2321 k 2 . 4P2 The solution is hc 1 T(,)uv k exp( d D )cosk E 1 ³ 2h (10.3.25) hhc 1 2 DEkkexp( d)sink exp( d),D 23³³21hhc fork z 0 , and hc P14 T(,)uv exp( d DEDE )[(k 2 d )k k ], (10.3.26) ³³21h 1 Pc 23 fork 0 . Here D uv , E uv and kii ,1,2, 3 , are real constants. C On the other hand, it results k 2 , and then the function (10.3.24) defines the 4 revolution surface (10.3.25). If we consider ON THE TZITZEICA SURFACES AND SOME RELATED PROBLEMS 281 3 3 UU tanh (C ) , V tanh V , (10.3.27) 2 1 2 the function h is written as 2UVcc huv(,) 1 Huv (,) 1. (10.3.28) ()UV 2 Therefore, the function H is a solution of the Liouville–Tzitzeica equation (10.2.6), which defines a ruled Tzitzeica surface. PROPOSITION 10.3.2. (Bâlă). The solution of Tzitzeica (10.3.25) is invariant under the transformation subgroup of G2 , for which the Lie algebra is generated, in the case ww of not ruled Tzitzeica surfaces, by ZC C . 12wwuv In the following we present the Shen method to construct a Weierstrass elliptic function Z(;2t 13 ,2Z ), by using the purely imaginary solution, Z1 and Z2 , of the Van der Pol’s differential equation 22 xxxkxtt P(1) t 0,, P k R, (10.3.29) with the initial condition x()tx0 0 . (10.3.30) Shen considers 1 P2 g g 2 , (10.3.31) 3 4 2 where g2 and g3 are the invariants of the Weierstrass equation (10.3.18) written under the standard form 23 4g2 g3 . (10.3.32) The solution requires that g2 and g3 must be real, and g3 must be negative. Shen showed that g2 is real if P and k are comparatively smaller than x0 . Introducing the iSZ3 quantity q exp , Shen showed also that g3 is negative for Z1 1 q ! . (10.3.33) 505 The quantities Z13 and Z may be computed by using the complete elliptic integrals in condition 16 0 Pg 4 . (10.3.34) 2 27 282 ON THE TZITZEICA SURFACES AND SOME RELATED PROBLEMS Shen proved that there exists always one and only one cycle on the phase plane for the solution. The equation (10.3.34) is quadratic and gives two real roots under the restriction imposed on x0 . One root is positive and the other is negative. The positive root leads to Shen’s solution. Ling has shown that there exists a second solution in which Z3 is no longer purely imaginary, whereas Z1 remains real, if the condition (10.3.34) is not satisfied. In this case we have 16 g 0 , or g !P4 . (10.3.35) 2 2 27 So, one of the three roots of the cubic equation remains real but the other two roots are complex conjugate. The negative root, which satisfies the first condition in (10.3.35) leads to a second solution. Ling introduces two invariants V4 and V6 in place of g2 and g3 11f g , V42 ¦ 4 60mn, f (2mnZ13 2 Z ) 1 f 1 g , (10.3.36) V63 ¦ 6 140mn, f (2mnZ13 2 Z ) where the summation omits the simultaneous zeros of mn and . From the variation of V4 and V6 it results that we are in the case when g2 and g3 are both negative, and the value of q is given by 1 qc i exp(S ) , 0 c . (10.3.37) 2 In this solution, the values of Z1 and q are found from 4 1 §·S 8448 g233 ¨¸()TTTT44, 12 ©¹Z1 6 1 §·S 44 44 g334 ¨¸(2TTT )(243T ) , (10.3.38) 432 ©¹Z1 where f n2 T3 12¦ q , n 1 f 2 12 (1)nnq . (10.3.39) T4 ¦ n 1 ON THE TZITZEICA SURFACES AND SOME RELATED PROBLEMS 283 Elimination of Z1 leads to an equation with real coefficients which can be solved numerically for a real root pq 2 exp( 2 S c ) , less than unity but greater than exp(S ) , for the specified g2 and g3 . Finally, Ling found iZ1 §·1 2lZ31 Z Sog¨¸. (10.3.40) S ©¹p If e1 is a real root of the cubic equation, the others two roots e2 and e3 are complex conjugate. Consider now the Liouville–Tzitzeica equation (10.2.6) and the Tzitzeica equation (10.2.8). If we denote ln h Z , these equations become Zuv exp Z, (10.3.41) and respectively Zuv exp Z exp( Z 2 ) . (10.3.42) THEOREM 10.3.2 (Bâlă) The generator vector field which describes the algebra of infinitesimal symmetries associated to the Liouville–Tzitzeica equation (10.3.41) are ww w Wfu () gv () [ fugvcc () ()] . (10.3.43) wwuv wZ THEOREM 10.3.3 (Bâlă) The vector fields which generate the Lie algebra of infinitesimal symmetries associated to the Tzitzeica equation (10.3.42) are ww w w Uu v, U , U . (10.3.44) 1 wwuv2 wu 3 wv If Z fuv(,) is a solution of the Tzitzeica equation (10.3.42), then the functions Z1 fu[exp( H ) , exp( H ) v ] , Z2 fu[, H v], Z3 fuv[, H ], whereHR , are also solutions of the equation. 10.4 The relation between the forced oscillator and a Tzitzeica curve Crăúmăreanu considers this example. Let consider the equation f ()tftgtH () (), (10.4.1) ForH 1 , (10.4.1) represents the forced oscillator equation. The driven function g()t is defined as 1 gt() , ab,R , a z 0 , H r1. (10.4.2) atH b 284 ON THE TZITZEICA SURFACES AND SOME RELATED PROBLEMS We can write (10.4.1) under the form 1 (at H b) . (10.4.3) Hft() ft () Differentiating (10.4.3) with respect to time, we obtain Hft() ft () aH . (10.4.4) [()()]Hft ft 2 Consider now in R3 a curve C in the form rrt (). This curve is called elliptic cylindrical if it has the expression rt( ) (cos t ,sin t , f ( t )) , fC f (R) . (10.4.5) The curve C is called hyperbolic cylindrical if it has the expression rt( ) (cosh t ,sinh t , f ( t )) , fC f (R) . (10.4.6) For an elliptic cylindrical curve, the torsion function W()t , and the distance from origin to the osculating plane dt() are defined by ffcccc W()t , (10.4.7) 1ffccc2 2 r()ffcc dt() . (10.4.8) 1ffccc2 2 For a hyperbolic cylindrical curve, the functions W()t and dt() are ffcccc W()t , (10.4.9) 1 (ffccc22 )(cosh 2 t sinh 2 tfftt ) 4ccc cosh sinh r()ffcc dt() . (10.4.10) 1 (ffccc22 )(cosh 2 t sinh 2 tfftt ) 4 ccc cosh sinh Consider now that the curve C is Tzitzeica with the constant a z 0 . For a cylindrical elliptic curve we have W()tftfcccc () ()t a , (10.4.11) dt2 () ( ft () ftcc ())2 and similarly or a hyperbolic cylindrical curve W()tftfc ()ccc ()t a . (10.4.12) dt2 () ( ft () ftcc ())2 The equation (10.4.11) is the same with (10.4.4) for H 1 , and the equation (10.4.12) is respectively the same with (10.4.4) for H 1. The solution of (10.4.4) for H 1 is given by ON THE TZITZEICA SURFACES AND SOME RELATED PROBLEMS 285 t sin(ut ) ft() f (0)cos t f (0)sin t ³ du, (10.4.13) 0 au b and the solution of (10.4.4) for H 1 by t sinh(ut ) ft( ) f (0)cosh t f (0)sinh t ³ du. (10.4.14) 0 au b PROPOSITION 10.4.1 An elliptic cylindrical curve rt( ) (cos t ,sin t , f ( t )) is a Tzitzeica curve, if and only ifft() is the solution of the forced harmonic oscillator 1 (10.4.13), with the driven force gt() , where the constanta is defined as atH b W()t a , and b is an arbitrary constant. dt2 () PROPOSITION 10.4.2 A hyperbolic cylindrical curve rt( ) (cosh t ,sinh t , f ( t )) is a Tzitzeica curve, if and only if (10.4.14) is verified. 10.5 Sound propagation in a nonlinear medium A chiral Cosserat material is isotropic with respect to coordinate rotations but not with respect to inversions. A chiral material has nine elastic constants in comparison to the six considered in the isotropic micropolar solids. Materials may exhibit chirality on the atomic scale, as in quartz and in biological molecules. Materials may also exhibit chirality on a larger scale, as in composites with helical or screw-shaped inclusions, solids containing twisted or spiraling fibers, in which one direction of twist or spiral predominate. Such materials can exhibit odd rank tensor properties such as piezoelectric response. Chiral materials include crystalline materials such as sugar which are chiral on an atomic scale, as well as naturally occurring porous material, bones, foams, cellular solids, honeycomb cell structures (Lakes). Consider the basic equations for an anisotropic Cosserat solid which is isotropic with respect to coordinate rotations but not with respect to inversions (Cosserat, Eringen). Balance of momentum VUkl,, ku l tt 0 . (10.5.1) Balance of moment of momentum mjrk, rH klr V lr UI k, tt 0 . (10.5.2) Balance of energy UH,,,t V kl()u l kt H klr I r t m kl I l, kt . (10.5.3) 286 ON THE TZITZEICA SURFACES AND SOME RELATED PROBLEMS The generalized Christensen’s constitutive law which describes the multi-relaxation processes in a chiral material is given by N ddiiVHf abmn mn Vkl¦¦ iklmn ii iklmn OHGrr kl ii 10ddtt (10.5.4) PNHNH(2 )kl klm (rCCC m IIGII m )1, r r kl 2 k , l 3, l k , N ddiim f I mcmn d mn kl¦¦ iklmn ii iklmn DIr,, rG klEI k lJI l, k ii 10ddtt (10.5.5) HGCCCCCr12332rr kl ()()(), Hkl Hklm m I m whereN is the number of different relaxation processes which are activated during the loading process, Vkl is the stress tensor which is asymmetric tensor, mkl is the couple stress tensor (or moment per unit area), H is the internal energy density, 1 H ()uu H( r I) is the strain tensor, u is the displacement vector, j is kl 2 k.,l l k klm m m microinertia, Hklm is the permutation symbol, and U the density of the material. The microrotation vector Ik refers to the rotation of points themselves, while the 1 macrorotation vector r Hu refers to the rotation associated with movement of kklm2 ml, nearby points. The coefficients aiklmniklmn ,b , ... are related to the material properties and to the deformation of the medium. The usual Einstein summation convention for repeated indices is used and the comma denotes differentiation with respect to spatial coordinates and a superposed dot indicates the time rate. We use rectangular coordinates xk (k 1, 2, 3 ) or ( x12 xx, yx, 3 z). In three dimensions there are nine independent elastic constants required to describe an anisotropic Cosserat elastic solid, that is Lamé elastic constants O , and P , ҏҏ the Cosseratҏҏ rotation modulus N , the Cosseratҏҏ rotation gradient moduli DE,,J , and Cii ,1,2, 30, the chiral elastic moduli associated with noncentrosymmetry. For Ci , the equations of isotropic micropolar elasticity are recovered. For D E J N 0 , equations (10.5.2) and (10.5.3) reduce to the constitutive equations of classical isotropic linear elasticity theory. From the requirement that the internal energy must be non- negative, we obtain restrictions on the micropolar elastic moduli 03dO 2PN, 02dPN, 0 dN, 03dDE J , JdEdJ, 0 dJ, and any positive or negative CCC123,,. Boundary conditions do not depend on assumed material symmetry. One may prescribe the displacements or the surface traction and the microrotations, or the surface couples on the surface. Consider, now, a slab of chiral material compressed in the z direction, and bounded by surfaces zH r (Lakes). Assume a monorelaxation process, and one component of displacement uz ()z , and one component of microrotation, Iz ()z to be nonzero. The motion equations (10.5.1) and (10.5.2) for stress p and the couple stress m are given by ON THE TZITZEICA SURFACES AND SOME RELATED PROBLEMS 287 (O 2 PN )(uuCCCz,,123,,zz WVV zzzt) ( )( Izzz W I zzzt)() Uuu ztt , WH zttt, , (10.5.6) 2N (1 WIWIKu2 )( u)()( UIWIj ) , (10.5.7) 0,z zzVV z , zzt DEJ z z,,, t z TTH z ttt whereWH , WV are the relaxation times under constant strains, and respectively constant stresses, and K0 with a coupling coefficient defined as ()CCC 2 K 2 123 . (10.5.8) 0 (2O PN )( DEJ ) For the case the relaxation times are zero, the motion equations (10.5.6) and (10.5.7) for stress p and the couple stress m are given by (2O PN )uCCCz,123,zz ( ) Iz zz U u z, tt , (10.5.9) 2N (1K 2 )u I UjI , (10.5.10) 0,z zz DEJ z z, tt with a coupling coefficient defined by (10.5.8). The traveling wave solutions of (10.5.9) and (10.5.10) can be found assuming [ zct , where c is the wave velocity. Thus, the above equations become 2 (2O PNUcu )zcc ( C123 C C ) Iz 0 , (10.5.11) 2N (1Ku2 ) cc IUIjc2 cc 0 , (10.5.12) 0 zzDEJ z where prime means the differentiation with respect to [ . The solutions for these equations are Iz I0 sinh p [, (10.5.13) 2(NO 2 PNUc2 ) p 222 , (10.5.14) ()(1)()(2DEJ K01CCC 2 3 U jc O PNUc) and CCC ue [ 123I[sinh p, (10.5.15) z O2 PNUc2 0 with ec , and I0 to be determined from the boundary and initial conditions. For the case the relaxation times are not zero, the dimensionless generalized equation of sound propagation in a chiral medium, written for the longitudinal displacement wave u , is 22 2 uuucuuufttD t E t [ xx J x G x ], (10.5.16) 288 ON THE TZITZEICA SURFACES AND SOME RELATED PROBLEMS involving the functions of nonlinearity D,,,,EJc G and f , whose specific expressions depend on u . With a proper change of function, the equation (10.5.6) may be reduced to ZZtt xx F(, ZZZx , t ), which plays an important role in the soliton theory. In particular, for F(Z ) sin Z, we have the sine-Gordon equation, for F()Z ZZ3 , we obtain an equation, which appears in solid state physics and high energy particle physics (Dodd et al.), and for 1 Z F()Z B (sin Z sin), the double sine-Gordon equation is obtained. 22 1 ForD J 0 , c 1, E G and f 4au 2 ,a ! 0 a constant, the equation u (10.5.16) becomes uu22 uutx 4au 2 . (10.5.17) tt uuxx By changing the function u expZ, we have ZZtt xx 4expa Z, and by changing the variables xt [, x t K, the above equation yields to the Liouville–Tzitzeica equation (10.3.41) Z[K a exp Z 0 . (10.5.18) 4 For f 4au2 b ,ab, ! 0 constants, the equation (10.5.16) yields the Tzitzeica u equation Eab(Z ) Z[K exp Z exp( Z 2 ) 0 . (10.5.19) Tzitzeica considers the linear system of equations (Tzitzeica) W[[ Ub [ W [ Oexp( U ) WK 0 , 1 WKK Ub K W K Oexp( U ) WK 0 , (10.5.20) W[K aU Wexp 0 , withOC , Oz0 , that are verified if U is a solution of (10.5.19), and W a function related to the ratio of two entire functions. The system (10.5.20) is invariant under the involution (Tzitzeica) 12WW (,exp,WOoUU ) ( , exp()[K ,O ). (10.5.21) W a W2 From (10.5.21) it results the Darboux transformation that leaves invariant the equations (10.5.20) ON THE TZITZEICA SURFACES AND SOME RELATED PROBLEMS 289 2 WW exp(Z ) exp(U ) [K, a W2 or 2 exp(Z ) ww log W exp(U ) , a [K whereZ and U are two different solutions of (10.5.19). By following the results of Musette and his coworkers, Conte and Verhoeven, we start with the Painlevé analysis of the Tzitzeica equation (10.5.19) with one family of movable singularities, in order to obtain the Bäcklund transformation of this equation. We check if the equation verifies the Painlevé condition. That means assume the existence of the transformation Z UD log W, E()Z 0, (10.5.22) withZ a solution of (10.5.19), D a singular part operator, and U an a priori unconstrained function. Then we choose the two, three, etc., order for the unknown Lax pair and represent this Lax pair by an equivalent Riccati pseudopotential YY12,,.... The equation (10.5.19) is invariant under the permutation (Lorentz transformation) (,ww[K ) oww (,) K[. (10.5.23) This invariance is conserved by the Lax pair. Since W is invariant under the permutation (10.5.23), we can define the unknown Lax pair on the basis (,,\\\xt) as §·\[ §fff123· ¨¸ ¨ ¸ ()wLLgg \ 0, g , (10.5.24) [K¨¸ ¨123¸ ¨¸ ¨ ¸ ©¹\ ©100¹ §·\[ §g123gg· ()M ¨¸ 0,Mhhh ¨ ¸ , (10.5.25) wKK¨¸ \ ¨123¸ ¨¸ ¨ ¸ ©¹\ ©010¹ with the link W \ . The nine coefficients fghjjj,,, j 1,2,3, are functions to be determined. The Riccati components Y12 and Y are defined as \ \ Y [ , Y K . 1 \ 2 \ The properties of the cross-derivative conditions are written as YY1,K[ 2, , ()()YYXXYXY1,[K1, K[ 0 1 1 2 2 0, (10.5.26) ()()YYXXYXY2,[K2, K[ 3 4 1 5 2 0 , 290 ON THE TZITZEICA SURFACES AND SOME RELATED PROBLEMS where 2 YYfYfY1,[ 1 1 1 2 2 f 3 , YY2,[ 1YgYgY 2 1 1 2 2 g 3 , 2 YYhYhY2,K 2 1 1 2 2 h 3 . The first relation (10.5.26) is identically verified, and the rest of relations (10.5.26) yield X j 0 ,j 0,1,...,5 , or Xfg03,3,13313323 t [ fgfgfhgg 0 , 2 Xf12,2,12213222 K[ g fgfgfgfh 0 , Xfg21,1,21213 K[ fhggg 0 , Xh33,3,32233113 [K g ghghfhgg 0 , Xh42,2,21213 [K g fhggg 0 , 2 Xghghghhgfh51,1,12213111 K[ 0 . The next step is to choose the link DflogW ( \ ) between the function W and the solution\ of a scalar Lax pair (10.5.24) and (10.5.25). For the link W \ , the truncation of the difference E()ZEU () is written as exp(Z ) DU log W exp( ) , (10.5.27) 33k EEfghUYY() ( , , , )kl 0. (10.5.28) Z ¦¦ kl j j j 12 kl 00 The relations (10.5.27) and (10.5.28) yield to ten equations EfghUkl(, j j ,,)0 j , kl, , (10.5.29) for determining the unknowns U and the nine coefficients fghjjj,,, j 1,2,3 . This process is repeated until a success occurs, namely |,w[KLM w |is equivalent to EU() 0 . Then, eliminating \ from the Darboux transformation 2 exp(Z ) ww log \ exp(U ) , (10.5.30) a [K and the scalar Lax pair, two equations for the Bäcklund transformation are obtained. Musette et al. have shown that the truncation of exp(Z ) has no solution. The truncation of exp(Z ) ON THE TZITZEICA SURFACES AND SOME RELATED PROBLEMS 291 exp(Z ) DU log W exp( ) , 2 EU() 0, D w w , (10.5.31) a [K generates four equations which admit some particular solutions of the form (10.3.23) and (10.3.24). According to the PROPOSITION 10.3.1, the solution (10.3.24) is a revolution ruled Tzitzeica surface. PROPOSITION 10.5.1 The solution in displacement of the sound propagation in a slab of a chiral material problem is represented by the revolution ruled Tzitzeica surface. 10.6 The pseudospherical reduction of a nonlinear problem Consider the one-dimensional problem of an anisentropic gas with its pseudospherical reduction given by Rogers and Schief in 1997. We present in this section the results of Rogers and Schief. The governing equations in a Lagrangian system of coordinates (,)X t are written as Ht vX , (10.6.1) U0vt VX . (10.6.2) The constitutive law is given by V ppX (, U ). (10.6.3) Here,p and U are the pressure and respectively, the density of the medium, U H 0 1 is the stretch, U is the density of the medium in the underformed state, and U 0 vXt(,) is the material velocity. The Lagrangian equations (10.6.1) and (10.6.2) and a constitutive law of the type V VH(.X ) may describe also the uniaxial deformation of nonlinear elastic materials with inhomogeneities. In terms of the Eulerian coordinates x xXt(,), we have d(1)ddx HXvt , (10.6.4) so that U0dddX Uxvt U . (10.6.5) In (10.6.5), X corresponds to the particle function \ of the Martin formulation. The independent variables are chosen to be p and \ , and we suppose U0 1. In this case we obtain the Martin formulation (Rogers and Schief 1997) of the Monge–Ampère equation as 2 [ ppp[[\\ \ Hp , (10.6.6) 292 ON THE TZITZEICA SURFACES AND SOME RELATED PROBLEMS where t [p , v [\ , d(x [\\\ [pp [ [p H)d \, (10.6.7) 0| [pp H f | . If a solution [\(,p ) of this equation is specified, then the particle trajectories are calculated from x [([[ [[ H )d\] , t [ , (10.6.8) ³ \\\pp p p in terms of p , for \ const. The isobars are parametrically expressed in terms of \ , forp const. By solving (10.6.8), the solution pt (\ , ) is obtained, and the original solution of (10.6.1)–(10.6.3) is determined in terms of the Lagrangian variables x xt(,)\ , vv (,)\ t, pp (,)\ t. (10.6.9) Rogers and Schief made the geometric connection of this problem. To show this, consider a surface 6 in the Monge parametrisation (1.7.7), whose Gaussian curvature is given by (1.7.9b). Let us introduce new independent variables p and \ pz x , \ zy , (10.6.10) and the dependent variable [ [ p x , [\ y . (10.6.11) Therefore, we have zyy zxx zxy [pp 2 , [\\ 2 , [p\ 2 . (10.6.12) zzxxyy z xy zzxxyy z xy zzxxyy z xy The Gaussian curvature (1.7.9b) yields 1 K 222 2. (10.6.13) (1\[[[p ) (pp \\p \ ) The Gaussian curvature may be set into correspondence with the Martin’s Monge– Ampère equation (10.6.6) by 1 H , (10.6.14) p .(1p22 \ ) 2 and UO22 . , (10.6.15) (1\p22 ) 2 ON THE TZITZEICA SURFACES AND SOME RELATED PROBLEMS 293 wp where O2 , with O the local speed of sound in gas. For the analog nonlinear wU |\ elastic problem, we have A2 . . (10.6.16) (1V22 X ) 2 wV where A2 , with A the Lagrangian wave velocity. The surface 6 is restricted to wH |X be pseudospherical, that is 1 K , a const. (10.6.17) a2 In this case the relation (10.6.16) gives 2 wV 2 22wV 22 (1VV!X ) |X 0 , V!0 . (10.6.18) wH |X a wH Integrating (10.6.18) with p oV and \ o X , we get aX22VV1 H [arctan( ) ]()D X , (10.6.19) 2(1XXX23/2 ) 1 2 1 V 2 2 with D()X arbitrary. For V|H 0 0() , it results D X =0. Now, we introduce 1 X 2 V 1tan[( X 2 cc )] , (10.6.20) a 0 into (10.6.19) and obtain 22 aXcc 0 121 H 2 [ ]sin( (cc 0 )) . (10.6.21) 2(1 X ) aa1 X 2 Relations (10.6.20) and (10.6.21) represent a parametric representation for the constitutive laws of the type V VH(,X ), for which the equations (10.6.1) and (10.6.2) are associated to a pseudospherical surface 6 . These equations lead to VXXtt H. (10.6.22) Using (10.6.14), we have a2 V [ V ] . (10.6.23) XX (1V222 X ) tt This equation has a solitonic behavior, if we take into account that for a pseudosperical surface 6 , the expression (1.7.12) of the Gauss curvature reduces to the sine-Gordon equation. 294 ON THE TZITZEICA SURFACES AND SOME RELATED PROBLEMS The mean curvature + of 6 given by (1.7.9a) is an invariant, and then we can write (1V22 )XX 2 V V (1 X ) V cot Z a vv t . (10.6.24) 2(1V223/2 X ) The map (,)Xto ( Xcc vt ,o t ),0| f vX | , (10.6.25) yields ww || v | , wwX tXtv t www |||| v , (10.6.26) wwwtvtX tX t v or ww1 ||tt , wwvvXt| X wwwvtX| ||vt |X . (10.6.27) wwwtvXtXt| From (10.6.24) it results 22 a 12V VXXvv VXt (1)( VX tX V ) cot Z a 223/2 . (10.6.28) 2(vXX 1)V The Gauss equations rrtv (, ) of the surface 6 becomes rzerzett tt333, tv tv Vv eXe t 3 , rzeXevv vv3 v 3 , (10.6.29) with the compatibility condition Vvt X . (10.6.30) This is a restatement of (10.6.2) with U0 1, by using VX |||,tXtvt v V vtX| X tv| , (10.6.31) vX |t in the condition 0| vX |f . The equation (10.6.1) is equivalent to the area preserving map expressed in terms of v and t independent variables ON THE TZITZEICA SURFACES AND SOME RELATED PROBLEMS 295 wH(,)X 1. (10.6.32) w(,)vt The expression of the Gaussian curvature (1.7.9b) becomes 222 VtvXX V vt .(1 V X ) , (10.6.33) and then wH(,)X wV .(1 V222 X ) | 1 . (10.6.34) ww(,)vt HX 1 Ifr is the position of a surface 6 with Gaussian curvature . , then (1.7.14) a2 or (1.7.15) hold. So, the construction of such pseudospherical surfaces yield Bäcklund transformation (1.7.16) for the sine-Gordon equation. In terms of the physical variables, the Bäcklund transformation becomes aXXcos] (c ) a cos] (VVc ) ttc , vvc , 1VVccXX 1VVccXX aXXcos]V (cc V ) zzcc V()(t t Xvvc ) , 1VVccXX 1VVccXX cos ], (10.6.35) 11V22 XX Vcc 2 2 where ddd,dddztXvztXv V ccccc V . (10.6.36) The Monge–Ampère equation 1 zz z22 (1 z z22 ) , (10.6.37) tt vv tv a2 t v determines the pseudosperical surface 6 , being invariant to the Bäcklund transformation (10.6.35). An important point of view is to interpret (10.6.35) in terms of the characteristic equations of (10.6.37) (Courant and Hilbert) aX aX D E aVD t , t , v , D 1V22 X E 1V22 X D 1V22 X aVE v , ztXvV 0 . (10.6.38) E 1V22 X DD D If ( tv,,,zX ,V ) is a solution of (10.6.38), then a solution of the Monge–Ampère equation (10.6.37) is z z((,),(,))D tv E tv , (10.6.39) 296 ON THE TZITZEICA SURFACES AND SOME RELATED PROBLEMS w(,tv ) obtained from the condition of a nonzero Jacobian . Therefore, the Bäcklund w(,)D E transformation (10.6.35) represents an integrable discretization of the nonlinear characteristic equations (10.6.38) of the Monge–Ampère equation. If we introduce the continuous variables Dˆ and Eˆ ˆ Dˆ H11n , E H22n , (10.6.40) for small Hi ,1,i 2 , we have the Taylor series O()2 , O()2 , (10.6.41) Z11 ZHZDˆ H1 Z22 ZHZEˆ H2 O(,)22. Z12 ZHZ 1Dˆˆ HZ 2ED HHZ 1 2ˆ E ˆ H 1 H 2 For PHH112 22 2 HHO(,)12, (10.6.42) P2 4aˆ the permutability theorem (1.7.23) reduces in the limit Hoi 0 , to the sine-Gordon equation 1 Z sin Z. DEˆ ˆ aˆ2 In terms of the physical variables, the Bäcklund transformation yields the discrete equations aXcos]11 (X ) tt1 , 1VV11 XX aXcos]22 (X ) tt2 , 1VV22 XX a cos]11 (VV ) vv1 , 1VV11 XX a cos]22 (VV ) vv2 , 1VV22 XX zz11 V()( ttXvv1 ) , zz22 V()( ttXvv2 ) , (10.6.43) and ON THE TZITZEICA SURFACES AND SOME RELATED PROBLEMS 297 1VV11 XX cos ]1 , (10.6.44) 22 22 11V XX V11 1VV22 XX cos]2 . (10.6.45) 22 22 11V XX V22 Using (10.6.42) we write 2 2 ]1111 HN O() H , ]222 SH N O() H2 , (10.6.46) with 1 P tan(] ) . 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This page intentionally left blank Index Active constitutive laws 203, 209 cardiac cycle 205, 211, 212, 213, 215, 216, 217 Active media 59, 70 218, 219 Active stress tensor 199, 201, 204, 208, 209 Cauchy theorem 15 Affine spheres 273 Cauchy–Riemann relations 39 centroaffinity 274 Causality principle 217 Amplification 59, 61, 214, 217 Cellular solids 285 Anisentropic gas 291 Chezy turbulent resisting force 242, 261 Anisotropic material 202, 215, 244, 285, 286 Characteristics 54, 85, 110, 166, 167, 220, 258 Autonomous system 83 Chirality 285 Chiral material (medium) 285, 286, 287, 290 Bäcklund transformation 39, 40, 41, 42, 43, 44, Chiral Cosserat material 285 78, 111, 113, 273, 289, 290, 295, 296 noncentrosymmetry 286 auto-Bäcklund transformation 40 Chiral elastic moduli 286 Bilinear form 25, 30 Christensen’s constitutive law 285 Biological molecules 285 Cosserat material 285, 286 Biological populations 49 Cnoidal Blood coefficients 236 functions 18, 19, 20, 139, 187 Blood flow 214, 220, 221, 235 method 17, 21, 23, 25, 185, 189, 190, 197, 209, hematocrit 220, 236, 240, 241 211, 247 microgyration 220, 239, 240, 241 solutions 1, 17, 78, 137, 138, 173, 185, 190, 206, peaking 221 285 pulmonary artery 214, 220 vibrations 173, 185, 187, 189, 211 steepening 221, 228, 229, 230 waves 18, 22, 23, 24, 25, 58, 146, 206, 242 thoracic aorta 230, 231, 232, 233, 234, 238, 239 Cole–Hopf transformation 110 Bond number 271 Conservation laws 53, 62, 66, 74, 78, 108 Bones 285 Constitutive laws 59, 206, 293 Bound-state 9, 11, 185, 189 active 203 Boussinesq equation 77, 107, 108, 153 passive 201 Breaking the waves 84 Continuity equation 87, 201, 216, 221, 256 Breather solution 119 Convergence criterion 250, 267 Brillouin zone 192 Coulomb wave functions 183, 185 Burgers equation 56, 74, 86, 107, 110, 111 Couple stress 242, 260, 262, 286, 287 Coupled pendulum 173, 175 Cantor-like structure 242, 243, 244 Crystalline material 285 admittance curve 252 Cantor set 243, 244 D’Alembert solution 54, 170, 172 fracton (localized vibration mode) 242, 243, 244, Darboux transformation 39, 288, 290 253 Darcy’s law 201 phonon (extended vibration mode) 243, 253 Dielectric constants 244, 245 piezoelectricity 244, 245 Dielectric wires 113 Cardiovascular system 198, 214, 215, 217, 218 Diffusion equation 75, 86, 110 adenosine triphosphate (ATP) 214, 215 diffusion coefficient 110 306 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS diffusion system 76 Froude number 271 reaction–diffusion 49 Fundamental solutions 5, 7, 8, 10, 12, 17, 90, 95, Dilaton 59, 61, 213 100, 101, 103 dilatonic mechanism 59 energy influx 198, 213, 214 Gaussian curvature 41, 273, 291, 292, 294 Dirac function 12 Gauss equation 294 Direct scattering problem 90 Gaussian function 57 Dispersion 55, 56, 59, 83, 85, 253 Genetic algorithm 197, 204, 207, 208, 209, 251, relations 21, 55, 56, 150, 156, 157, 158, 193, 225 252, 266 Dissipation 56, 59, 110, 154 Green–Lagrange strain tensor 199, 201 equations 56, 58, 86, 113, 114 Group of symmetry 53, 62, 63, 64, 65, 66, 273, DNA 121 274, 275, 276, 277 DRIP media 149, 159, 160, 163 infinitesimal symmetries 65, 277, 283 Dynamic viscosity 261 Group velocity 55, 56, 150 Eigenfunction 2, 4, 96 Hamiltonian 67, 74, 78, 85, 108, 110 Eigenvalue 2, 3, 35, 38, 96, 228, 248 Hamilton principle 126 problem 248, 249, 251, 252 Heat equation 56, 110 Eigenfrequency 249, 251, 252 Heaviside function 12 Elastica 121, 143, 145, 146, 152, 191 Helical inclusions 296 Elastic potential 174, 201, 246 Helical rod 149, 155, 157, 158 Elastic constants 54, 55, 159, 244, 285, 286 Helmholtz conditions 69, 70, 72 third-order 244 Hemodynamics 220 Electric field 244, 245, 247, 249, 252 Heterogeneous string 149, 159, 166, 168 potential 244 Hirota method 25, 29 induction vector 244 Hirota operator 25 Elliptic cylindrical curve 284 Honeycomb cell 296 Elliptic integrals 18, 19, 23, 118, 138, 139, 141, Hydrodynamic resistance 216 143, 147, 148, 281 Hyperbolic cylindrical curve 284, 285 Elliptic Weierstrass function 20, 187, 273, 281 Hyperbolic functions 1, 18, 122, 143, 152 Epoxy resin 243, 244, 251, 253 Hypergeometric series 90, 91 Equipartition of energy 107 Error 24, 250 Interaction of waves 102, 163, 242, 258, 259 Euler angles 122, 123, 126, 128, 130, 131, 132, Internal energy 59, 248, 249, 286 139, 143, 147, 148, 149, 155 Inverse Lagrange problem 69 Euler function 91, 92, 93 Inverse scattering problem 2, 3, 12, 17, 30, 85, 94, Euler–Lagrange equations 53, 66, 69, 255 105, 173, 228 Eulerian coordinates 59, 122, 123, 198, 205, 291 Inverse algorithm 207, 249, 250 Evolution equations 1, 25, 29, 43, 44, 53, 62, 73, Involution 288 74, 75, 78, 198 Irrotational motion 86, 269 Extended mode of vibration (phonon) 242, 244, Isobars 291 253, 255 Isotropic energy function 199, 201 Faddeev condition 5, 13 Jacobian 295 Fermi, Pasta and Ulam problem 82, 107, 159, 173, Jacobi elliptic functions 1, 18, 19, 20, 21, 58, 81, 191 110, 173 Fitness function 207, 208, 250, 266, 267 Jacobi elliptic integrals 18 Fliess expansions 34 Jacobi matrix 21, 63 Flow velocity 220, 221, 224, 228, 229, 234, 235, Jacobi multiplier 70 238, 239, 240, 241 Jordan lemma 15 Flux 84, 216 Foams 160, 285 Kadomtsev and Petviashvili equation (K-P) 30 Focusing of waves 59 Kinetic energy 129, 130, 173 Forced oscillator 283, 284 Kink 42, 51, 113, 152, 153, 154, 155 Fourier transform 12, 14 anti-kink 153 Fractal 243 Klein–Gordon equation 56 Fracton (localized mode of vibration) 242, 244, Konno–Ito method 105, 153 252, 253, 255 Korteweg–de Vries equation 3, 5, 18, 21, 25, 30, Frequency 55, 60, 146, 157, 193, 216, 231, 234, 44, 53, 55, 58, 61, 75, 79, 84, 89, 94, 105, 107, 237, 242, 245, 249, 253 110, 153, 227 INDEX 307 Krishnan solution 187 Morse interaction force 192 Lagrange equations 53, 66, 67, 69, 127, 131, 174 Navier–Stokes equation 221, 260 Lagrangian 66, 67, 68, 69, 70, 72, 130, 131 Nijenhuis operator (hereditary operator) 74 coordinates 134, 198, 205, 290 111, 228 Green strain 202 Noether theorem 53, 66, 67, 68 multipliers 69, 70, 72 Nonlinear diffusion equation 75, 86, 110 Lamé constants 55, 286 Nonlinear partial differential equations 1, 21, 132 Lamb waves 245 Laplace equation 39 Nonlinear Schrödinger equation 77 Lax pairs 85, 288, 289, 290 Normal mode 107, 252, 253, 254 Laurent series 50, 51 Left ventricle 197, 198, 199, 205, 207, 209, 211, Objective function 207, 208, 249, 250 214, 216, 220 Open channels 242, 259 cardiac cycle 205, 211, 212, 213, 215, 216, 218 Ordinary differential equations 69, 128 endocardium 197, 200, 204 Orthonormal basis 122 epicardium 197, 200, 204, 209 heart 220, 229, 230 Painlevé equations 48 intramyocardial pressure 211, 213 Painlevé property 1, 2, 46, 47, 48, 49, 51, 52, 113, myocardial tissue 200, 201, 215 288 myocardial deformation 197 Passive constitutive laws 201 porosity 199 Pendulum 72, 111, 115, 117, 151, 173, 174, 175, sarcomere length 200, 204, 208 191 super ellipsoid surface 198 Periodic boundary conditions 21, 82 Legendre equations 7 Periodic solutions 105, 109, 110, 111, 191, 193, Legendre function 93, 97 238, 266 Lewis method 168 Periodic structures 191, 193, 242, 253 LEM solutions 34, 35, 173, 177, 180, 183, 190 Permeability tensor 199 Lie algebra 65, 276, 281, 283 Permutability theorem of Bianchi 43, 44, 45, 111, Lie derivative 33, 34 296 Lie transformation 80. 89 Perturbation method 30, 121 Linear equivalence method (LEM) 31, 32, 35, 37, Phase portrait 239, 240, 241 190 Phase shift 102, 103, 106, 153, 259 Liouville–Tzitzeica equation 76, 274, 277, 278, Phase velocity 23, 55, 56, 230, 269 281, 282, 288 Phonon (extended mode of vibration) 242, 244, 253, 255 Localized waves 1, 5, 9, 53, 78, 82, 83, 122, 173, Piola–Kirchhoff stress tensor 199, 200, 201, 203 193 Piezoelectric material 242, 243, 244, 246, 248, Localized mode of vibration (fracton) 242, 244, 251, 253, 285 252, 253, 255 Piezoelectric constants 244, 245 Lorentz transformation 288 Poincaré map 218 Potential Burgers equation 75 Marchenko equations 17, 98, 101, 104 Potential energy 201, 203, 248 Martin formulation 291, 292 Potential function 2, 5, 9, 12, 13, 17, 90, 96, 99 Material symmetry 286 Potential Korteweg–de Vries equation 74, 75 Maxwell stress tensor 245 Pre-hilbertian space 3 McLaurin series 107. Pressure waves 221, 232, 233 Mean curvature 40, 293 Pseudo potential energy 214 Micropolar coefficients 260, 267, 286 Pulsatile flow 220, 228, 235 Micropolar elastic moduli 286 Micropolar model 220, 235, 242, 259, 285 Quantum mechanics 192 Microrotation 235, 242, 259, 260, 269, 286 Microinertia 236, 286 Recursion operators 73, 74, 75 Miura transformation 85, 109 Reflection coefficient 7, 8, 9, 17, 90, 92, 93, 171 Modal interacting 191 Relaxation function 200, 201 intermodal interaction 242 time 215, 216, 286, 287 Modified Korteweg- de Vries equation 74 Repulsive energy function 203 Monge representation of the surface 40, 291 repulsive range function 203 Monge–Ampère equation 291, 292, 295 Residuum theorem 16 Monoclinic symmetry 244 Riccati equations 47, 49 308 INTRODUCTION TO SOLITON THEORY: APPLICATIONS TO MECHANICS Riccati pseudopotential 288, 289 Theta function 18, 21, 23, 173, 197, 247, 251 Rotation vector 259, 304 Thin elastic rod 121, 122, 125, 128, 130, 132, 149 Runge–Kutta method 216 Toda lattice 191, 192, 242, 299 Toda–Yoneyama equation 173, 194, 196 Saphenous artery 221 Torque spring 111 Scaling transformation 73, 74 Torsion function 284 Schrödinger equation 2, 5, 12, 90, 94, 99, 100 Torsional motion 150 Screw-shaped inclusions 285 Transmission coefficient 7, 8, 9, 11, 15, 93, 94, 95 Screw-symmetric operator 169 Tricomi system 115 Self similarity 243 Troesch problem 37 Seymour–Varley method 149, 160, 161, 163 Turbulent flow 242 Shen–Ling method 20, 273, 281 Tzitzeica equation 64, 76, 273, 274, 275, 276, Symmetry transformation 67, 74, 75, 84 279, 280, 281, 282, 283, 288 Simple pendulum 72, 115, 117, 151 Tzitzeica curves 293, 284, 285 Sine-Gordon equation 41, 42, 43, 76, 111, 112, Tzitzeica surfaces 64, 273, 274, 275, 277, 278, 113, 293, 294, 296 280, 281, 290 Singular points 46, 47 Solitary waves 54, 78, 79, 82, 242, 255, 259, 269, Ursell number 23 270, 272 Soliton 1, 13, 30, 31, 53, 58, 61, 78, 79, 82, 83, Van der Pol’s equation 20, 273, 281 102, 105, 106, 113, 149, 152, 173, 191, 193, 228, Variational principles 69, 248 230, 233, 234, 235, 272, 278 Vibrations 107, 149, 156, 166, 168 anti-soliton 153, 154, 217 cnoidal 173, 185, 187, 189, 211 asymmetric 61, 198 Coulomb 183 solutions 1, 18, 27, 28, 29, 30, 50, 62, 78, 121, fracton 253 138, 149, 192, 232, 293 free 249 collision (interaction) 105, 153, 154, 193 linear 149, 150 multi-soliton solution (N-soliton solution) 29, 94, nonlinear 149, 173 100, 103, 113, 193, 228, 266, 269 phonon 253 two-soliton solution 43, 45, 83, 100, 102, 105, torsional 149 106, 109, 111, 113, 119, 220, 228, 229, 233, 239, transverse 54, 149, 155, 157, 158, 159 241 Viscosity 261 theory 1, 39, 78, 121, 220, 235 of blood 247 Sound propagation 285, 287, 290 Volterra integral equations 6, 115 Spiraling fibers 285 Strain energy function 199, 202, 203 Waves 3, 7, 11, 18, 21, 30, 54, 56, 59, 76, 79, 82, Stratified fluid 255 84, 102, 105, 113, 153, 160, 163, 191, 238, 242, String equation 12, 53, 54, 121, 149, 159, 166 249, 258, 267, 272, 287 Sturm–Liouville theory 2, 3 cnoidal 18, 22, 23, 24, 25, 27, 58, 146, 206, 239, Subgroup 65, 276, 277, 281 242, 266 Subalgebra 277 harmonic 55, 56 Super ellipsoid surface 198 linear 53, 55, 114 Superposition principle nonlinear 58, 78, 192, 214 linear 121, 154 of translation 79 nonlinear 1, 24, 25, 42, 43, 173, 185, 187, 206, shock 84, 110, 111 211, 242 Wavelength 55, 60, 84, 88, 267 Surface tension 242, 269, 272 Wave equation 31, 59, 108, 159 Synge method 149, 166 Wave number 21, 55, 60, 189, 193, 247, 266 Weierstrass equation 21, 23, 49, 58, 81, 110, 135, Taylor series 34, 66, 225, 295 173, 185, 186, 187, 279, 281 Tensors Wronskian 8, 9 asymmetric stress 242 deformation gradient 199 Yang functional 248 Cauchy stress 199 Yih analysis 242, 255 couple stress 242, 260, 262, 286, 287 Yomosa theory 221, 231 Maxwell stress 245 Yoneyama solution 173, 194, 196 Piola–Kirchhoff stress 199, 200, 201, 203 Young's elastic modulus 125, 236 Thermodynamic pressure 259 Fundamental Theories of Physics Series Editor: Alwyn van der Merwe, University of Denver, USA 1. M. Sachs: General Relativity and Matter. A Spinor Field Theory from Fermis to Light-Years. With a Foreword by C. Kilmister. 1982 ISBN 90-277-1381-2 2. G.H. Duffey: A Development of Quantum Mechanics. Based on Symmetry Considerations. 1985 ISBN 90-277-1587-4 3. S. Diner, D. Fargue, G. Lochak and F. Selleri (eds.): The Wave-Particle Dualism.ATributeto Louis de Broglie on his 90th Birthday. 1984 ISBN 90-277-1664-1 4. E. Prugovecki:ˇ Stochastic Quantum Mechanics and Quantum Spacetime. A Consistent Unific- ation of Relativity and Quantum Theory based on Stochastic Spaces. 1984; 2nd printing 1986 ISBN 90-277-1617-X 5. D. Hestenes and G. Sobczyk: Clifford Algebra to Geometric Calculus. A Unified Language for Mathematics and Physics. 1984 ISBN 90-277-1673-0; Pb (1987) 90-277-2561-6 6. P. Exner: Open Quantum Systems and Feynman Integrals. 1985 ISBN 90-277-1678-1 7. L. Mayants: The Enigma of Probability and Physics. 1984 ISBN 90-277-1674-9 8. E. Tocaci: Relativistic Mechanics, Time and Inertia. 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