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Introduction to Soliton Theory: Applications to Mechanics Fundamental Theories of Physics

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Introduction to Soliton Theory: Applications to Mechanics Fundamental Theories of Physics 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 ©2005 Springer Science + Business Media, Inc. Print ©2004 Kluwer Academic Publishers Dordrecht All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Springer's eBookstore at: http://ebooks.kluweronline.com and the Springer Global Website Online at: http://www.springeronline.com Contents 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.
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