Appendix A Linear and Nonlinear Mathematical Physics: from Harmonic Waves to Solitons
It does not say in the Bible that the fundamental equations of physics must be linear. -E. Fermi 1
In this section we describe briefly a very impressing development of Nonlinear Science strongly connected with the discovery of solitons ( Manevitch, 1996). In the last few decades, problems of a qualitatively new type have appeared in different fields of physics. A turning point was transition from quasi-linear (almost linear) Physics to essentially nonlinear Physics. Such a transition would be impossible or, at least, near hopelessly difficult without a parallel development of mathematical techniques fitting well with the new physical ideas. Therefore it is very important that modern progress in physics took place at the same time as amazing mathematical discoveries. Similar key notions (though in different guises), for the first time, after a long period, turn out to be at the center of attention simultaneously of physicists and mathemati• cians. Certainly a soliton or particle-like wave is a case in point. This term first appeared in the pages of scientific journals some forty years ago and one sees it, perhaps, in all fields of physics. The role of the soliton in nonlinear mathematical physics is similar to the role of har• monic oscillations and waves (corresponding, for example to pure tones in acoustics and pure colors in optics) in the quasi-linear case. Therefore we can symbolically characterize the path from quasi-linear to nonlin• ear mathematical physics, as it is called in the title of this section, as: 183 184 ASYMPTOTOLOGY: IDEAS, METHODS, AND APPLICATIONS from harmonic waves to solitons. We will attempt to describe this path briefly, comparing the physical content of the new and the conventional notions. We will present examples of new the possibilities opening up in physics due to the discovery of solitons.
1. THE QUASI-LINEAR WORLD Two main problems existed in theoretical physics from its very be• ginning at the end of the 17th century: discovery of fundamental laws of nature and derivation of those consequences that admit experimental testing. The second problem is in essence a subject of mathematical physics in the wide sense of the word. The laws of nature are natu• rally formulated in mathematical language, as a rule, in the form of differential equations. The corresponding equations (of those laws) in celestial mechanics, historically the first field of theoretical physics, are already nonlinear because the force of gravitational attraction between planets is not a linear function of their mutual distance but is inversely proportional to the squared distance. Unfortunately, as distinct from the case of linear systems, there are no general methods of analytical solution for such nonlinear problems. Therefore the fate of mathematical physics could be under threat from the very beginning . This threat did not been materialize because of two main reasons. First of all, the initial object of mathematical physics was the Earth-Moon system which can be considered autonomically. Thus, the simplest case arose: the two-body problem. The problem of motion of the Earth around the Sun, in first approximation, can also be reduced to the two-body problem by neglecting the effects of other planets. On the other side, the system of differential equations of celestial me• chanics has properties, connected with of space-time symmetry, which are called conservation laws. Thus, the homogeneity of time causes con• servation of full energy, the homogeneity of space-conservation of mo• mentum, the isotropy of space (equivalence of all directions )-conservation of angular momentum. Every conservation law provides a certain rela• tion between the sought-for functions which can be used for decreasing the dimensionality of the problem. In the two-body problem the conser• vation laws mentioned above are sufficient for the derivation of laws of motion of the planets as guessed by Kepler. However, these symmetries of space-time are already insufficient for a full analytical treatment of the three-body problem. This means that despite the universality of natural laws, there are no equally universal techniques for their theo• retical analysis. To attain a true correspondence between natural laws and techniques for their analysis, it is necessary to find more essential restrictions than the ones dictated by space-time symmetry. Appendix A: Linear and Nonlinear Mathematical Physics 185
The development of linear mathematical physics (or mathematical physics in the narrow sense of the word) has played a decisive role in this field. This development was strongly related with development of acoustics, the theory of elasticity, the theory of heat and mass trans• fer, and optics. Linearization turned out to be a very efficient universal procedure. Its efficiency has both mathematical and physical founda• tions. From the physical point of view, linearization is a consequence of the general principle: the reaction to a certain action is proportional to this action (as a rule, it is a good approximation for actions of weak intensity). From the mathematical point of view linearization gives a possibilities of full solutions to the problem due to appearance of an appropriate number of additional internal symmetries and, as a conse• quence, new relations between variables. It is possible to find such a change of variables that splits the system of linear equations into in• dependent equations. In the dynamical case each of these equations describes an elementary or cooperative motion: harmonic oscillation or an harmonic wave preserving its spatial form. The complicated behavior of the system is arises from combinations (superposition) of a number of elementary motions. The validity of the superposition principle is one of the most important consequences of linearization. The change of variables mentioned above means a transition to "col• lective" coordinates, describing special non-localized motions--normal modes, depending on internal properties of the system and conditions at the boundaries. Every normal mode corresponds to a harmonic oscil• lation or wave. Their combinations with coefficients depending on the initial conditions and external forces give information about behavior of every particle. This turns out to be more close to the essence of the problem, because the particles (for example, atoms in a solid) interact strongly with each other. However, the collective modes are indepen• dent (in a quasi-linear system they are almost independent). Here it is worth mentioning that at the microscopic level the language of particles is, naturally, quite adequate when interaction is absent (ideal gas) or weak (real gas). For solids the language of waves replaces the language of particles. The universality of linear mathematical physics showed up in that similar equations turned out to be fundamental for different fields of physics. For equilibrium problems (statics of deformed solid, electro• statics, and so on) the Laplace equation appears frequently in one or another form. In linear dynamics (acoustics, theory of elasticity, op• tics) the d' Alembert equation is the fundamental one; in relaxation pro• cesses (heat conductivity, diffusion) the Fourier equation plays such a role. These three equations constitute the basis of linear mathematical 186 ASYMPTOTOLOGY: IDEAS, METHODS, AND APPLICATIONS
physics. This strongly effected the formation of mathematical thinking and development of mathematical notions (spectra of natural frequen• cies, Fourier series and integral, Green function, and so on) as far as up to the second half of the 19th century. As in Newton's time, scientists, who were mathematicians and physi• cists simultaneously, created mathematical physics. While this takes place, the problems of theoretical physics constitute the most impor• tant part of mathematical studies. Later, 'internal problems' become more and more significant for mathematics; its demarcation from physics starts being be noticeable. In the development of mathematics, ideas from non-Euclidean geometry, group theory, and set theory played cru• cial roles. At the same time, in physics the role of the concepts of thermodynamic equilibrium became very essential. This circle of ideas suggests that the motion of the particles of gas, liquid, and solid is fully chaotic at the microscopic level. A fundamental problem that has be• come one of the main subjects of statistical physics has arisen: how to relate macroscopic properties of systems with microscopic dynamics? It has been discovered that to explain the thermal properties (for exam• ple, thermal capacities) it is sufficient to calculate the simplest types of motion: free motion of particles (gas) and small oscillations (solid) in the framework of linear dynamics. Complicated problems of nonlinear dynamics became less essential for physicists, especially as the new elec• tromagnetic theory of Maxwell, that provides the unification of optic, electric and magnetic phenomena, was a linear one, and so was Fresnel's optics. The discovery of quantum mechanics showed that linearity and the superposition principle are fundamental laws of Nature. The pure (sta• tionary) states of the system here are the analogs of normal modes. An arbitrary state can be found as a superposition of pure states. A certain infinite combination is connected with a unique particle. We see that the field of efficient applications of linear mathematical physics is rather wide. However the picture of the world created by it does not reflect in certain cases the essential features of experimentally observed phenomena. Thus, the dependence of normal frequencies on amplitudes, abrupt changes of amplitude for small shifts of excitation frequencies, and the excitation of self-sustained oscillations are observed in experiments. There are other deviations from the behavior predicted by linear models; for example, thermal expansion of solids. It turns out, however, that in the cases mentioned above the nonlinear effects can be considered as "small" despite the possibility of qualitative changes in the behavior of the system; "smallness" in this context means a possibility to Appendix A: Linear and Nonlinear Mathematical Physics 187
calculate all effects mentioned within the framework of the quasi-linear approach, considering the linear theory as a first approximation. As a result, an extremely wide picture of (calculable) processes and phenomena appeared which can be called the quasi-linear picture of the world. Despite its apparent diversity, there are simple building bricks in its foundation: non-interacting particles or normal modes. Linear mathematical physics, certainly, is the basis of this quasi-linear world. However, being so universal and deep, the quasi-linear approach and lin• ear mathematical physics are insufficient for understanding some very important phenomena and regularities. Elucidation of reasons for this insufficiency and looking for new approaches turn out to be strongly connected with discovery and investigation of solitons and with the for• mation of nonlinear mathematical physics.
2. ON THE WAY TO NONLINEAR PHYSICS As we mentioned above, theoretical physics from the very beginning dealt with nonlinear problems. However, the term "nonlinear mathe• matical physics" has appeared in our days only. The point is that up to recent time there were no central notions in nonlinear problems sim• ilar to the normal modes and the superposition principle in the linear case which could provide a unified point of view and high efficiency for nonlinear physics. The soliton is one such notion. The uncompleted history of ideas connected with this term starts with an observation described by the English engineer J.S. Russell more than 150 years ago. He observed when a ship in the Edinburgh-Glasgow channel stopped rather abruptly "the birth 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 diminu• ation of speed." The language of this paper, unusual for us to hear, pre• serves the living surprise of the naturalist observing a rare and unusual phenomenon. What is the reason for this surprise? We mentioned above that in the theory of wave motions of small intensity (linear wave theory) the simplest motions are infinite sinusoidal harmonic waves. Their pro• file does not change in time, and dissipation of energy (if it is present) simply leads to a gradual decrease of amplitude. The speed of such waves depends, as a rule, on their length (this property is called dis• persion). This absence of interaction as well as the dependence of wave speed on its amplitude is also characteristic of linear waves. It is pos• sible to create a perturbation of arbitrary complexity (even a solitary wave) by combining harmonic waves. However every profile besides the harmonics themselves spreads out in time (Fig. A.l) because of the dif• ference of speeds for harmonic waves of different length (dispersion). 188 ASYMPTOTOLOGY: IDEAS, METHODS, AND APPLICATIONS
u /r
D
Fzgure kl. Spreading of wave packet due to dispersion, 11 is displacement of parti• cles, x is spatial coordinate, t is time.
This effect also manifests itself even in full absence of dissipation. Grav• itational waves in water as observed by Russell are characterized also by the property of dispersion. It is a reason why even well-known scientists supposed that the wave which seems purely a raise is a part of usual wave with alternating regions of raise and decrease. It was not easy to understand the nature of a solitary wave. It was concealed in complicated equations of hydrodynamics, and obviously some taking into account of nonlinearity is required here (in the opposite case dispersion spreading is inevitable). However, nonlinearity without accounting for dispersion would lead to breaking (overturning) of the solitary wave due to the dependence of its speed on the amplitude of displacement. The first theoretical studies of the solitary wave appeared in the 1870s. However, the paper of two Dutch physicists published in 1895 exerted the most noticeable influence on the subsequent events. In this paper the famous Korteweg-de Vries equation (KdV-equation) was first obtained and solved for certain particular cases. Initially, the KdV-equation was considered as a certain approximation for equations of hydrodynamics, valid for a thin layer of an ideal incompressible liquid (the approxima• tion of "shallow water"). However, by now the fundamental importance of this equation in physics is obvious. In the KdV-equation, dispersion and nonlinearity are taken into account in a first approximation (major mathematical difficulties are connected with accounting for nonlinear• ity). Therefore, it seems that the trend to spreading out and the trend towards breaking (overturning) of the solitary wave, already mentioned above, impel us to doubt its prolonged existence. However, solutions of Appendix A: Linear and Nonlinear Mathematical Physics 189
0 X
Figure A.2. Conservation of the profile of a solitary wave due to mutual compensa• tion of dispersion and nonlinearity. the KdV-equation gave evidence that these two destabilizing effects can compensate each other and thus see to the conservation of the profile of a solitary wave (Fig. A.2). Let us give an intermediate summary. The solitary wave in hydrody• namics looked like an exotic phenomenon requiring specific conditions for its realization and decay due to different perturbations (for example, as a result of the inevitable collision of two waves of different amplitudes because of difference in speed). A total change of the initial opinion concerning solitary waves and the birth of the term "soliton" turned out to be connected with a problem from other field of physics. We have already mentioned that beginning of the second half of the 19th century, thermodynamics and its foundations, statistical physics, became leading fields of physics. It has been already shown that for the prediction, for example, of the thermal capacity of a crystal, consideration of linear harmonic waves as elementary excitations is quite sufficient. To make a thermodynamic description applicable, a certain "chaos" (uniform distribution of energy between all degrees of freedom) has to exist. This requires interaction of elementary motions. In our case they are collective ones. However, the mutual independence of harmonic waves, corresponding to normal modes of linear theory, means that the amount of energy, given initially to each of such modes, is preserved (for that mode) for an indefinitely long time. Consequently, a transfer of energy to a uniform distribution among all normal modes (thermalization) has to be credited to the influence of nonlinearity. It has to be small enough in order to not change the thermal capacities noticeably. This problem was the object of a numerical simulation per- 190 ASYMPTOTOLOGY: IDEAS, METHODS, AND APPLICATIONS formed in Los Alamos by E. Fermi, J.R. Pasta, and S.M. Ulam in the 1950s, on one of the first computers. At the beginning of this experiment its authors did not doubt that the case in point was not the effect of ther• malization but just the determination of the rate at which it would take place. However, the mathematical model of a one-dimensional crystal consisting of 64 particles coupled by weakly nonlinear springs behaved in a quite unexpected manner. Instead of thermalization of the energy, given initially to one of the normal modes, what happened was some• thing quite different: a quasi-periodic transfer of energy between several modes. This work gave rise to a series of analogous studies and the collection of questions raised by them was called the FPU problem. In 1967, N.J. Zabusky showed that the equations of motion for a one• dimensional crystal like the one studied in the mathematical experiment described above reduce in the weakly nonlinear and long wave length approximation to the KdV-equation. So, an analogy between two rather different fields of physics was discovered. Simultaneously, new insights in the FPU problem were obtained: indeed, the KdV-equation has solitary waves as its solutions! In the numerical experiment it was possible to specify conditions different from those dictated by an exact particular solution for a solitary wave (for example, the conditions providing a collision of two such waves). Contrary to predictions, the collision did not lead to the destruction of the solitary waves. Moreover, the profiles of the waves involved and their speeds were the same after collision (Fig. A.3). The only trace the interaction left was a certain shift in the wave phase after the collision. Another observation was not less surprising: when an initial perturbation was of a rather general form, its evolution led to a decay in the form of a series of solitary waves. In other words, an initial profile quite different from a solitary wave can be considered as an instantaneous picture at a moment of interaction of a series of solitary waves traveling with speeds depending on their amplitudes. Earlier, such a simple behavior was known for particles only. Let us note that all regularities mentioned above except the phase shifts were noted, as it became clear later, by Russell in special experiments with solitary waves in water. However, he had not realized the analogy of these waves with particles. This understanding the results of the numerical experiments inspired Zabusky and Kruskal (1965) to introduce a special term, "soliton", for the solitary waves which are solutions of the KdV-equation. This term comes from "solitary wave", but the ending "on" is meant to suggest the particle-like behavior of solitons. The appearance of the notion "soliton" meant that a first stable synthesis of wave and particle in the framework Appendix A: Linear and Nonlinear Mathematical Physics 191
u
0 X
Figure A.3. Collision of solitons. of classical physics was obtained. The work of Kruskal and Zabusky led to complete change of meaning of the role of solitary waves in Physics. It has called forth intensive analytical studies, because the properties of the solitons of the KdV-equation turned out to be very unusual. However, the successes that followed seemed incredible. It turned out that the KdV-equation has deep internal symmetry, as represented by the pres• ence of an infinite number of conservation laws. In 1967 C.S. Gardner, J.M. Greene, M.D. Kruskal, and R.M. Miura showed how it is possible to obtain general solutions of the KdV-equation. These comprise a wide class of initial conditions including as a very particular case the solitary wave. Even now this discovery is still amazing: till very recently general solutions for only a few nonlinear problems of low dimensionality were known. Here the problem at issue was a system of infinite dimension• ality described by nonlinear equation with partial derivatives. The idea behind the solution was very amazing also. It was (and is) based on a correspondence between the KdV-equation and a certain linear problem. The inverse transformation to the solution of the initial equation was the most difficult stage. As it happened this inverse problem had been studied earlier by I.M. Gelfand, B.M. Levitan, and V.A. Marchenko. 192 ASYMPTOTOLOGY: IDEAS, METHODS, AND APPLICATIONS u
0 X
Figure A.4. The envelope soliton describes the spatial localization of particles' oscil• lations and moves with constant velocity.
It is a famous inverse scattering problem. In 1971 V.E. Zakharov and L.D. Faddeev proved that the KdV-equation can be considered as an infinite-dimensional analog of the exactly solvable problems of classical mechanics. It usually happens that deep physical ideas and notions become a fo• cus of interaction of, seemingly, distant fields of mathematics. This hap• pened in the case of solitons to the fullest extent. Their study restored interest in all kinds of classical parts of mathematics and stimulated the development of a number of new, frequently very abstract, directions of investigation. These new areas of interest play an important role in modern mathematics. Intensive and constructive mathematical studies created a basis for deeper insight of solitons in physics. As was mentioned above, equations which have soliton-like solutions turn up in a wide circle of the problems of theoretical physics. Some of the applications of the KdV-equation to hydrodynamics and solid state physics were mentioned above. This equation appears in those cases when both dispersion and nonlinear• ity are small. In the case of strong dispersion and weak nonlinearity the Nonlinear Schrodinger equation (NSE), which has envelope solitons (Fig. A.4) as solutions, plays an analogous role. Its general solution, containing the envelope soliton as a particular case, was obtained by V.E. Zakharov and A.B. Shabat in 1971. The nonlinearity can be of such a kind that a system has two or more equilibrium states, in the simplest case with equal potential energy. The best known exactly solvable equation, describing the dynamics of such systems, is the so-called sine-Gordon equation. This equation has nu- Appendix A: Linear and Nonlinear Mathematical Physics 193
X
Figure A.5. Kink, soliton-like solution of the sine-Gordon equation. Instant profile of a wave. merous applications in many fields of Solid State Physics, Nonlinear Optics, and Particle Physics. The simplest soliton-like solutions de• scribing transitions between two neighbor equilibrium states, is called a kink (Fig. A.5). Not all types of solitons can be obtained using a quasi-linear approach, for example, in the assumption that a linearized system is considered as a first approximation. In this sense the equations giving rise to solitons as well as their solutions are essentially nonlinear. The number of soli• ton equations is still increasing and their solutions lead to new physical ideas, connected with the particle-like properties of solitons. The univer• sality of soliton equations allows us to talk about the birth of Nonlinear Mathematical Physics and a new synthesis of Physics and Mathematics. As a result, fundamental new possibilities appear for the explanation and prediction of phenomena which cannot be explained or predicted in any other manner. 3. HOW SOLITONS WORK What is the physical content of these new kinds of objects, that are the subject of Nonlinear Mathematical Physics? First of all, as is seen from the above, that there other elementary (i.e., persistent in time) ex- 194 ASYMPTOTOLOGY: IDEAS, METHODS, AND APPLICATIONS citations than collective normal modes (and harmonic waves) and that they can also be localized modes. Harmonic waves of infinite prolon• gation lead to changes of phase in which oscillating particles or points of a field occur but do not transfer energy. To realize a transfer of en• ergy in the linear case it is necessary to create a group of waves forming a so-called wave packet which has initially the form of a single pulse. In contrast to a wave packet, which inevitably destroys itself in time due to dispersion, the soliton is a stable formation and, consequently, provides a most efficient mechanism of energy transfer. Moreover, its ve• locity can exceed the speed of sound (in the medium concerned), which is the maximal propagation velocity for linear waves. The existence of this mechanism is confirmed not only by such grandiose phenomena as tsunamis or waterspouts (tornado). Its important role is also manifest from the analysis of a whole series of processes in solids, biological and artificial polymeric molecules, optic fibers, and other systems. One such process is heat conductivity. Despite numerous studies, its microscopic mechanism in non-metal solids is not quite clear to date. Therefore it is difficult to derive the macroscopic equation of the heat transfer (Fourier equation) which was mentioned above from data relat• ing to the structure of crystal and the potentials of atomic interactions. The FPU problem has already shown that a nonlinearity does not lead automatically to chaotic motion, which is a necessary prerequisite for a classical law of heat transfer. For the existence of normal heat conductivity and the validity of the Fourier law at a macroscopic level, factors are important which coun• teract the formation and motion of solitons (if the role of latter ones is decisive the heat conductivity has to be practically infinite). This means that solitons have to be destroyed over time. The rate of such destruc• tion is one factor determining the applicability or non-applicability of the Fourier equation. Only a few properties of a solid, such as thermal capacity or elas• ticity, can be explained by supposing that the solid is an ideal lattice, with atoms or molecules oscillating with small amplitudes near their equilibrium states. So, the yield stress and ultimate strength are essen• tially lower than the values predicted by the theory of an ideal lattice. Therefore, the physicists have been forced to introduce such notions as structural defects, such as vacancy, dislocations, cracks, and others even before the experimental techniques for their discovery had appeared. The main property of a structural defect is: in the region where it is localized the lattice is strongly transformed. Besides, to be responsible for various different physical processes, these defects have to be highly enough mobile. Solitons satisfy both these conditions. Their role man- Appendix A: Linear and Nonlinear Mathematical Physics 195
ifests itself especially directly in crystalline polymers, i.e., in ordered solids formed by long and flexible macromolecules. In this case the in• tramolecular bonds are essentially stronger than those between neighbor macromolecules, and quasi-one-dimensional models are fully applicable. On the other side numerous nonlinear effects caused by the flexibility of the chains and the presence of many equilibrium configurations turn up Let us consider a chain of strongly coupled monomers (groups of atoms), the left part of which is in one state and the right is in other state with localized transition region (it is supposed that the system has at least two homogeneous equilibrium states). The displacements of the monomers form a kink, similar to the solution of the sine-Gordon equation mentioned above. A soliton of this type was first obtained by Ya.I. Frenkel and T.A. Kontorova when modeling line (i.e., local• ized along a line) structural defects in crystals. It models the sim• plest defect of such a type: a dislocation. Let us note that in this case the different homogeneous equilibrium states correspond to a shear in the chain direction exactly over an integer number of interatomic dis• tances. The equations describing structural defects in polymer crystals are more complicated, but they have also soliton-like solutions which correspond to two-dimensional (domain walls), line (dislocations), and zero-dimensional defects (vacancies). So, there is a possibility to avoid introducing structural defects "manually" in the model of a solid but to obtain them directly as solutions of corresponding equations for an ideal structure. At the same time, it turns out that such a kink can move with a constant speed, and this motion can be considered as a "transfer of state" along the chain. A similar mechanism is more profitable from the energy point of view than a simultaneous transition of an entire chain to a new state. Therefore, the mobility of kinks provides them with an important role in those processes where, along with energy transfer, a "state transfer" (plasticity, polarization, magnetic phenomena) can occur. The history of nonlinear Mathematical Physics, which is full of un• expected twists and turns, is interesting and instructive in many ways. From the mathematical point of view, the discovery of the soliton and a realization of the roles it can play could have already happened in the 19th century. The history of the soliton and Nonlinear Mathematical Physics is rather illogical but illustrates the deep inevitability of the in• teractions of Physics and Mathematics. We discussed very schematically only certain stages of this history and just a few applications of solitons, mainly to quasi-one-dimensional systems, similar to polymer crystals and biological macromolecules. The recent physical theories claiming a 196 ASYMPTOTOLOGY· IDEAS, METHODS, AND APPLICATIONS construction of a unified theory of fundamental interactions turn out to be essentially nonlinear, contrary to the classical electromagnetic the• ory A hypothesis that elementary particles are multi-dimensional soli• tons has been put forward by many physicists. Looking for such solitons is one of the significant directions of research in modern Theoretical Physics.
Notes 1 cited as in Ulam, 1960, p. 19. Appendix B Certain Mathematical Notions of Catastrophe Theory
1. REPRESENTATION OF FUNCTIONS BY JETS Let us discuss briefly the mathematical notions which turn out to be essential in connection with the problems of Catastrophe Theory. In the example considered (in Chapter 3) we dealt with transcendent functions and their power series expansions in a certain point. As this takes place we kept the first terms of these expansions only. So, the analysis concentrated on the local behavior of the functions and it was good enough for the prediction of such behavior with a good accuracy. The above representation (replacement of the function by a sum of the first terms of its power series expansion) is strongly connected with the mathematical notion of jets. If the derivatives of all orders exist for a function y = j(x1, x2, ... , xn) with respect to all variables in a certain point which can be identified with a coordinate origin, we associate (at that point) to the function y its formal Taylor expansion (Poston and Stewart, 1978). Then the k-th jet of the function y = j(x1, x2, ... , Xn) is the sum of first k + 1 terms of its expansion (the k-th term is a sum of all order k terms). For example, for a function of two variables y = f (x 1, x2) the first jet is of the form:
·l aj aj J J = J(o,o) +-a I x1 +-a I x2 (B.l) X1 0 X2 0
197 198 ASYMPTOTOLOGY: IDEAS, METHODS, AND APPLICATIONS
a / /
y=sln(x)
/
\~ f ''
Figure B.i. Representation of function y = sin x in the point x = 0 by its jets: a - y = l (sin x ), b - y = j 3 (sin x), c- y = j 5 (sin x), d- y = /(sin x ), e - y = j 9 (sin x), f- y = j 1 1(sinx), g- y = l3(sinx). and the second jet:
The value R = f -lf is called a "tail" of expansion. By way of illustration, he changes in the k-th jet as the label k increases for the function y = sinx is shown in Fig. B.l. In Catastrophe Theory a main topic is the possibility of the replace• ment of a function with certain of its jets so as to preserve a correct de• scription of the qualitative peculiarities of its local behavior near a point under consideration. In contrast, in Calculus the analogous question for functions with convergent power series expansions conventionally has been considered from the quantitative side (Maurin, 1989). Convergence of the series, i.e., the possibility of infinite corrections to the quantitative description of a function due to increase of the number of power terms, is determined by the tail (terms with large indexes). Meanwhile, the quali• tative properties of the function in the vicinity of a certain point depend on the first terms of expansion. Just how many first terms are needed Appendix B: Certain Mathematical Notions of Catastrophe Theory 199
is a question which can be answered using the notion of equivalence of a function and its jets. 2. EQUIVALENCY OF A FUNCTION AND ITS K-TH JET When replacing a function with the first terms of its power series ex• pansion we have to be sure that they are equivalent. But what does that mean exactly? Let us suppose that there is a transformation of the function into its k-th jet which does not lead to any additional singular• ities besides the ones inherent in the function. Such a transformation is symbolically denoted by T, so that the k-th jet is a result of its action on function f(x): Tf(x) = l J(x), where x is the set of variables x1, x2, ... , Xn· The action of transfor• mation Ton the function f(x) is realized (by definition) as a (possibly nonlinear) coordinate transformation
Tf(x) = J(Tx). (B.2)
If, for example, y = f(x) =ax, (a= const) and the transformation T is squaring the variable x: x' = Tx = x2 , then T f = a(Tx) = ax' = ax2 . Along with transformation T of a function into its k-th jet, we consider the inverse transformation: (B.3)
The relationships (B.2) and (B.3) should be considered as the equa• tions for determination of the transformations T and r-1 . Their solu• tions exist for certain values of k depending on the function f (x) and the expansion point. If the function f(x) is equivalent to its first jet, that is, there is a transformation converting this function to its first jet, then f(x) is certainly equivalent to all its jets with indices k > 1 also. Such a function is called 1-defined function and the first jet is called a minimal jet. If the function is not equivalent to the first jet but is equivalent to its second jet, it is called 2-defined function etc. Quite generally, a function is said to be finitely determined if it is equivalent to one of its finite order jets. Not all functions are finitely determined. So, a specific classification of functions arises. It permits us to formulate what are the simplest local representations for the class of functions under consider• ation. This representation should embody all the qualitative properties (peculiarities) of the behavior of the function behavior near the point considered. In this sense we can get exact answers to questions concern• ing the local behavior of the function by studying a much more simple 200 ASYMPTOTOLOGY: IDEAS, METHODS, AND APPLICATIONS object: a certain one of its jets. The problem under discussion is not considered part of Conventional Calculus with its quantitative approach but demands the use of ideas and methods from topology. Therefore, we will mention only certain facts which are necessary for understanding the results of Catastrophe Theory. First question is: how to find the transformations T and r-1? It turns out that for a function y = f(x) with a convergent power series expan• sion all such transformations can be obtained in terms of power series expansions in the independent variables, containing the linear terms (al• though the expansions of the functions can include every power). More• over, such transformations are reversible. Therefore, the search for T and r-1 can be easily put in algorithmic form and can be performed using modern software. Usually, it is easier to find the inverse transfor• mation r-1, i.e., to solve the equation r-1jk f(x) = j(x). Let us prove, by way of illustration, that the function y = f(x) = a1x + a2x2 + · · ·, where a 1 -/= 0 is equivalent to its first jet. Looking for an inverse trans• formation we have to solve the equation r-1p f(x) = f(x). Because 1 1 r-1j 1j(x) = j 1j(T-1x) = j 1j(x ) = a1x , the problem is reduced to 1 finding a function X ( x) converting the relationship
(B.4) into the identity. Let us try to find the solution as an power series expansion I 2 x =fJ1x+f32x + .... (B.5) After substitution of this expression in equation (B.4) we obtain
a1 (f3Ix + fJ2x 2 + · · ·) = a1x + a2x2 + · · · and find that fJ1 = 1, fJ2 = a2/ a1, .... So, the transformation (B.3) allows one to recover the entire function just knowing its first jet and consequently the function considered is 1-defined. Equivalency of the function to its first jet (if the latter one is non• zero) happens in the case of any number of variables. For example, if the function of two variables is of the form j(x1,x2) = a1x1 + a2x2 + b1xi + b2x~ (a1 -/= 0), the transformation y-l can be found as
x I1 =XI + al1 ( b1x 21 + b2x22) ;
However a simple example (Maurin, 1989) shows that such a con• clusion (equivalency of function to its first non-zero jet) is not valid Appendix B: Certain Mathematical Notions of Catastrophe Theory 201
for the functions of more than one variable if its power expansion be• gins, for example, with the terms of order two. Let y = !1 (x1, xz) = xf + x~ +xi+ x~ + · · ·, where "· · ·" corresponds to terms of higher order. The condition of equivalency of the function to its first non-zero (i.e., second) jet has the form:
T-1j 2.fl = T-1 (xf) = x? =xi+ x~ +xi+ x~. Searching for a solution of this equation as
I 2 2 X1 = f31x1 + f32X2 + /33Xl + f34X1X2 + {35x2 + · · ·
(for the sake of simplicity we do not write the transformation for x~) we write
Equating the coefficients corresponding to the same powers on the left and right sides of this equation we find /31 = 1, /32 = 0. But we cannot arrange for equality of cubic coefficients. This means that this function is not 2-defined. Actually, it is easy to show that it is a 3-defined function. 3. REPRESENTATION OF FUNCTIONS BY JETS IN ORDINARY POINTS The case of a non-zero first jet in a point considered is the simplest one, because the transition to the first jet allows us to obtain a linear function instead of a nonlinear function of many variables. Moreover, in this case our function is equivalent to a linear function depending on one variable only. For example, in the case of the function y = j(x1, x2), considered above, after replacing it with its first jet j 1(x1,x2) = a1x1 + a2x2 we 1 can find an elementary transformation T1- replacing the independent variable x1 with the first jet a1x1 + azxz. The corresponding equation for determination T!1 has the form
(B.7)
Searching for the solution of (B.7) in a form x~ = ')'1x1 + 1'2x2 we obtain that 11 = a1, 1'2 = a2. Consequently the initial function is equivalent to a linear function depending on one variable x~ only. This conclusion is valid for the functions of any number of variables with non• zero first jet in the point under consideration. Such points are called ordinary ones. From a geometrical point of view, a complicated surface can be replaced near an ordinary point by a many-dimensional plane (in the case of two variables-with a plane in the standard sense, i.e. a 202 ASYMPTOTOLOGY: IDEAS, METHODS, AND APPLICATIONS
.f'
Figure B.2. Replacing function y = x~ +x~ at the point (1, 1) with its first non-zero jet (plane). two-dimensional plane {Fig. B.2)). However, as we saw above a special role in the applications of Catastrophe Theory is played by the critical points in which the partial derivatives of the function with respect to all variables are equal to zero, so that, consequently, the first jet is zero. 4. JETS AT NON-DEGENERATE CRITICAL POINTS The point x = 0 is critical for all three functions considered above (y = x2, y = x3, y = x 4 ). However small additional terms do not change the qualitative behavior of the function in the first case only. It turns out that a function of one variable with zero first jet but non-zero second one is equivalent to its second jet. A form of this is also valid for functions of many variables (Morse's theorem) if the determinant of the second derivatives matrix is not equal to zero at the point x = 0. All such critical points are called non-degenerate ones. For the function y = x2 the second derivative at the point x=O is equal to 2, so this point is non-degenerate critical. A function of n variables in a non-degenerate critical point is equivalent to the sum of n squares of variables. In this sum l coefficients are equal to 1 and n - l coefficients are equal to -1. Append·ix B: Certain Mathematical Notions of Catastrophe Theory 203
Figure B.3. Replacing function y = xi - x~ +xi + x~ at the point (0, 0) with its first non-zero jet (hyperboloid).
From a geometrical point of view it means that the function is locally equivalent to a many-dimensional saddle surface (if n = 2, l = 1 it is a two-dimensional saddle (Fig. B.3), if n = 2, l = 0 it is like a paraboloid (Fig. B.4)). As in the particular case y = x2 , small perturbations do not qualitatively change the behavior of a function at a non-degenerate critical point; the only thing that happens is that the position of this point is shifted. As for the functions y = x3 and y = x4 the condition presented above is not satisfied and consequently the point x = 0 is degenerate. Degener• ate points are of the greatest interest from the viewpoint of qualitative change of behavior of different systems. However because small per• turbations can remove degeneration, functions with degenerate critical points can model real processes in Nature only if they enter into the problem as a set of functions depending on certain control parameters. In such a case, degeneration can often be realized for certain values of the control parameters. If all the functions in the family have a simulta• neous critical point and the dependence on the parameters is essential in an appropriate sense, there always is a (possibly complex) set of values of the control parameters that makes the critical point degenerate. 204 ASYMPTOTOLOGY: IDEAS, METHODS, AND APPLICATIONS
Figure B.4. Replacing function y = x? + x~ +xi+ x~ at the point (0, 0) with its first non-zero jet (paraboloid).
5. JETS AT DEGENERATE CRITICAL POINTS A power expansion of a function of n variables at a degenerate point is not equivalent to its second jet, i.e., to a sum of squares with coef• ficients of all the variables because at least one coefficient is equal to zero. Such a situation corresponds to certain combinations of control parameters. In general, critical points can be once, twice etc. degen• erate. It turns out that at a once-degenerate critical point, a function of n variables is equivalent to the sum of n - 1 quadratic terms and a term containing the variable, which is absent in this sum, in power more than two (Arnol'd, 1994, Poston and Stewart, 1978). For a function of one variable this means that the quadratic term is simply absent in the expansion, as is the case for the functions y = x3 and y = x 4 . Cer• tainly, the once-degenerate case will be the most frequently occurring because a change of control parameters, as a rule, leads to the disap• pearance of just one coefficient corresponding to quadratic terms. The case of twice-degenerate points has to be more rare. It corresponds to zero values of two coefficients at once. The two cases just mentioned are the most important for applications of Catastrophe Theory, because Appendix B: Certain Mathematical Notions of Catastrophe Theory 205
it is very difficult to arrange for threefold-degeneration. Also, for once• and twice-degenerate points we can find standard functions of one and two "degenerate" variables which have to be added to sums of n - 1 or n - 2 quadratic variables to obtain a jet which is equivalent to the function considered in the degenerate critical point. In the first case these functions are third (or fourth, fifth etc.) powers of the degenerate variable. More frequently only the quadratic term is absent. Then we obtain the "fold catastrophe" discussed above. A second important case occurs when both coefficients, corresponding to second and third powers of the degenerate variable are equal to zero. This case corresponds to "cusp catastrophe". When this takes place, the function of n variables is equivalent to a sum of n- 1 quadratic terms and a third or fourth power of the unique degenerate variable. In the case of a twice-degenerate point, a minimal jet consists of n -- 2 quadratic terms and two additional terms of higher order. 6. CONTROL PARAMETERS We emphasized above that degeneration of critical points in models describing physical, chemical or other systems can be realized as a re• sult of change of control parameters. Their number may be rather large. Therefore the objective of Catastrophe Theory is not only to reveal the "dangerous" (degenerating) variables, the real number of which is equal usually to unity or two. It is necessary also to find a minimal number of control parameters determining the qualitative changes in the behavior of the system under consideration. One simplification of such a prob• lem is the natural assumption that only "small perturbations" of the function at a degenerate critical point need to be considered. If a func• tion together with its small perturbations is considered it is termed a deformed function. A small deformation does not qualitatively change the behavior of a function at an ordinary or non-degenerate critical point. Actually, in such cases the terms of higher degree than contained in a minimal jet of the undeformed function can be eliminated by appropriate an transformation T. It is easy to verify that this is the case for functions of one or two variables. As to the additional terms of first and second degree in a minimal jet of ordinary or non-degenerate critical points, they do not change the structure of a minimal jet. A simple analysis (Poston and Stewart, 1978) shows that in the case of degenerate critical points the contributions of perturbations with re• spect to non-degenerating coordinates as well as to products of degener• ating and non-degenerating coordinates can be considered as negligibly small. At the same time the perturbations with respect to degenerating 206 ASYMPTOTOLOGY: IDEAS, METHODS, AND APPLICATIONS coordinates containing higher powers than those in a minimal jet may be excluded also by appropriate transformation. So, only the pertur• bations of lower powers remain, and the corresponding coefficients are "effective" control parameters, which are equal to zero at the critical point. This means that the principal constituents of minimal jets with respect to degenerating variables are always the terms of third, fourth or higher degree mentioned in the previous section. Because of this, for a deformed function, we have to take into account only the powers of degenerating variables smaller than minimal ones at the critical point. Let us note that from all of the small perturbations present we should consider the perturbing terms of first or second degree as the principal ones that change the structure of a minimal jet and, consequently, the type of the point. Certainly, it is hardly reasonable to expect that the program described above can be realized for arbitrary multiplicities of degeneration and for an arbitrary number of essential control parameters. However, if the multiplicity of degeneration is equal to one or two, it is possible to deduce the well-known list consisting of seven catastrophes (Arnol'd, 1994, Poston and Stewart, 1978). For each of these catastrophes the minimal jet depends on one or two variables and no more than four parameters. The remaining variables enter in the minimal jet as a sum of quadratic terms. Because stationary states correspond to extrema of functions y = f(x), their determination can be reduced in the cases under consideration to one or two nonlinear algebraic equations with respect to degenerating variables and to a system of n- 1 or n - 2 linear homogeneous equations with respect to the remaining variables having a trivial (zero) solution. Consequently, the Catastrophe Theory leads to colossal simplifica• tions of the initial problem due to a determination of a small number of "dangerous" variables. Their behavior is described by a small num• ber of nonlinear algebraic equations containing a few essential control parameters. Appendix C Asymptotics and Scaling Transformations
Alexander D. Shamrovsky
Alexander D. Shamrovsky is currently Professor of Applied Mathematics at the Zaporozhie State Technological Academy, Ukraine.
Here we consider the "Descartes folium" equation in detail, using coordinate and parameter scaling transformations as well as power series expansions (Shamrovsky, 1997). Let us begin with the equation
X 3 + Y3 - kXY = 0 ( C.l) with an arbitrary value of k. After the transformations
X=Jf31 X* ' Y==6f3 2 Y* , k=Jf33 k* (C.2) the terms of ( C.l) have powers of 6 as coefficients. Write down the cor• responding exponents preserving the same order of terms as in equation (C.l). This gives (C.3) An invariance condition guaranteeing preservation of the form of equa• tion (C.l) can be written as follows:
(C.4)
This is possible if (C.5) with an arbitrary f. So, equation (C.l) admits the transformations
X = P X*, Y == PY*, k = P k* . (C.6)
Now introduce new variables X y (C.7) X= k' y = k 207 208 ASYMPTOTOLOGY: IDEAS, METHODS, AND APPLICATIONS
which are invariants of the transformations (C.6) and find the equation
(C.S)
which is a starting point for asymptotic analysis. Performing the transformations
(C.9)
we obtain new coefficients of (C.S) which are powers of 15 with exponents
3o:, 3(3, 0: + (3 . ( C.lO)
Simultaneous equality of all these exponents is possible only if o: = (3 = 0. It means that equation (C.S) does not admit any more transformations of the type under consideration, preserving its invariance (besides trivial ones). This is quite natural thing to happen because we passed already to invariants of the transformations (C.6), preserving the form of (C.S). Supposing further that 15 < 1 we will assume that
x*c-vl, y*c-vl. ( C.ll)
1. ESTIMATION OF VARIABLES Let us consider the cases of maximal simplification. I. 3o: = 3(3 ::=; o: + (3, which means o: = (3 ::=; 0. The list (C.lO) takes a form 3o:, 3o:, 2o: . (C.12) The minimal (in absolute value) values of o: and (3, which make the numbers in the list (C.12) integral, are o: = (3 = -1, so this list takes the final form -3, -3, -2 (C.l3) and, consequently, x "" y > 1. The corresponding simplified equation can be written as (C.14) This equation is invariant with respect to some new transformations
( C.l5)
This allows us to introduce a new variable z = y / x which is an invariant of these transformations (C.15). Then equation (C.14) takes the form
z 3 + 1 = 0. ( C.16) Appendix C: Asymptotics and Scaling Transformations 209
II. Let us consider now the second case 3a = a + {3 ::=; 3{3 i.e. {3 = 2a ~ 0. Instead of the list (C.10) we now have
3a, 6a, 3a. (C.17)
The smallest values of a and {3, making these exponents integral are a= 1/3 and {3 = 2/3. Then y < x < 1. The simplified equation can be written as x 3 - xy = 0. (C.18) It has an additional admitting transformation of the form
x = [Jlf3x*, y = [J2f3y*. (C.19)
The corresponding invariant is z = y / x2 , and we can write the simplified equation (C.18) as z-1=0. (C.20)
III. The final case to consider is 3{3 = a + {3 ::=; 3a i.e. a = 2{3 ~ 0. Instead of the list (C.10) we this time have
6a, 3a, 3a. (C.21)
The smallest integer values for (C.21) correspond to a = 1/3 and {3 = 1/6, so this list takes the form
2, 1, 1 (C.22) and, consequently, x < y < 1. The simplified equation can be written in the form y3 - xy = 0. (C.23) Taking into account the new transformations
x = [J2/3x*, y = 8lf3y* (C.24) we obtain the corresponding invariant z = y2 jx and equation (C.23) takes the form z-1=0. (C.25) 2. SUBSEQUENT APPROXIMATIONS Above we have obtained the equations of initial approximation (max• imal simplification) for all three cases. To solve things represent x and y in the form of series
00 00 X= I:Xi, y = LYi. (C.26) i=l i=l 210 ASYMPTOTOLOGY: IDEAS, METHODS, AND APPLICATIONS
Along with the transformations (C.9) carry out the transformations
( C.27)
where i = 1, 2, ... , so that the series (C.26) take the form
00 00 x = oa L:x?o 1- 1, y = 0/3 LY~ 0 i-1 . (C.28) i=l i=l Let us assume again that 6 < 1 and substitute the series (C.28) into equation (C.8):
00 ex; 00 a3a I: xibi-l I: x; oi-l I: x'k ok-1 z=l r=l k=l
(C.29)
00 00 -oa+B L x.;n om-1 L y~ On-1 = 0. m=l n=l Now consider the concrete values a and f3 found above. I. a = f3 = -1. Removing the common factor o- 3 we have
00 00 00 00 00 00 L L L x';:rj xk oi+i+k-3 + L L L Yi Yj Yk 5 i+J+k-3 i==l j=l k=1 i==1 j==l k==l (C.30) ex; 00 -5 L LX~ y~ om+n-2 = 0. m=l n=l
Looking at equal powers of 5 in (C.30) leads to the equations
:L 1:; :rj x'k + L y: yj Yk - :L :r~ y~ = 0 , ( C.31) i+j+k=p i+j+kc.cp m+n+2=p where p = 3, 4,.... The value p denotes a order of approximation, the sum under every summation sign runs over the all possible values of indices, corresponding to the given p. Let us note that in the third sum in (C.31) there are no possible values rn and n which could corre• spond to the value p = 3. Therefore, this sum does not affect the first approximation. The infinite system of equations (C.31) is invariant with respect to the transformations (C.27), corresponding to a= f3 = -1,
( C.32) Appendix C: Asymptotics and Scaling Transformations 211
where i = 1, 2, .... This makes it possible to rewrite the equations (C.31) in the initial variables without change of form:
I: XiXjXk + I: Yi Yj Yk- I: Xm Yn = 0, (C.33) i+j+k=p i+j+k=p m+n+2=p where p = 3, 4, ... Thus, both the equations (C.33) and the series (C.26) do not contain the formal small parameter 8. Let us consider the equations of the first several approximations:
p= 3: xr + yf = 0'
p=4:
p= 5:
p = 6:
-X3 Yl - X2 Y2 = 0 . (C.34) The transformations (C.32) for these four equations have the form
.1"-1 * Yl = u Y1, Y2 = Y2, (C.35) X3 = 8xj, Y3 = oyj, The presence of these transformations allows us to decrease by 1 the number of variables to consider after the transition to invariants
(C.36)
Then the equations (C.34) take the form
z3 + 1 = 0,
3hx + 3x~ + 3zhy + 3zy~ - Y2 - x2z = 0 ,
3I4x + 6x2hx +X~ + 3I4y + 6y2hy + Y~ - hy/ Z - f3xZ- X2Y2 = 0 . (C.37) The recurrent system of equations (C.37) can be easily solved (first find z, i.e., YI(xi), then y2(x2) and so on). However we can simplify the 212 ASYMPTOTOLOGY: IDEAS, METHODS, AND APPLICATIONS solution further remembering that we actually searching for the depen• dence y(x). This means that we can consider the variable x as given. So. we can suppose that :r 1 = x, x2 = x 3 = X4 = 0. Then we have from (C.37), taking into account (C.36),
z 3 + 1 = 0,
3zy2- 1 = 0,
3zhy + 3zy§- Y2 = 0,
And we obtain
z = -1, Y2 = -1/3, hy = 0, l4y = 1/81 and 1 1 Y1=-x, Y2=-3, Y3=0, Y4 = 81x2 . The solution of equation (C.8), taking into account four approxima• tion steps, takes the form 1 1 y = -x - 3 + 81x2 · The first two terms on the right side give the equation of the asymp• tote and the third one describes the deviation of the solution from the asymptote for large x. II. a= 1/3, j3 = 2/3. Removing the common multiplier o we have
00 00 00 00 00 00 L L L xjx*x'ko i+i+k-3 + o 2:: 2:: 2:: Yi y*y'k8 i+j+k-3 . . k J . . J z=l J=l =1 z=l J=l k=l (C.38) 00 00 - 2:: 2:: x:ny~ om+n-2 = o. m=l n=l Equating in (C.38) the coefficients of equal powers of 0 leads to (after returning to the initial variables)
L XiXjXk+ L YiYjYk- L XmYn=O, (C.39) i+j+k=p i+j+k+l=p m+n+l=p where p = 3, 4, .... The infinite system of equations (C.39) is invariant with respect to the transformations following from (C.27) for a = 1/3, j3 = 2/3, X .= oi-2/3x* y. = oi-1/3y* z z ' z z ' ( C.40) Appendix C: Asymptotics and Scaling Transformations 213 where i = 1, 2, ... The equations (C.39) in explicit form (for the first few values of p) are p= 3: xf- X1Y1 = 0,
p=4:
p = 5:
p= 6:
(C.41) Now the transformations (C.40) have the form
_ d/3x* _ s:4/3x* X 1 - u ll Y1 - us:2/3y* ll X 2- u 2,
The invariants of these transformations are z=yl/xi, hx=x2/xi, l2y=y2/xf, hx=x3jxi, (C.42) hy = Y3/x~, l4x = x4jx~ 0 , l4y = Y4/xF . The equations (C.41) in invariant form are z-1=0,
( C.43)
-l2xhy- l3xhy- zl4x = 0 ·
Assume, just like before, that x1 = x, x2 = x3 = X4 = 0, we obtain from (C.43), using (C.42), z-1=0, 214 ASYMPTOTOLOGY: IDEAS, METHODS, AND APPLICATIONS
Then z = 1, hy = z3 = 1, hy = 3z2l2y = 3, l4y = 3z2hy+3ziiy = 12. The solution of equation (C.8), incorporating the first four approxima• tions, can be written as
III. o: = 1/3, (3 = 1/6. Repeating the calculations of case II we obtain in this case Appendix D Asymptotic Approaches: Attempt at a Definition
Rem G. Barantsev
Rem G. Barantsev is currently Professor of Hydro- and Aerodynamics at the Saint• Petersburg State University, Russia.
lei bas la verite ne nous sera jamais connue dans son ensemble ... L 'idee de l 'eternite est incompatible avec l 'existence de tout etre sujet a variations et a successions. -Augustin-Louis Cauchy 1
1. ASYMPTOTIC METHODS OR A NEW MATHEMATICS? The first time that one encounters something like asymptotics is in school geometry with the idea of an asymptote defined as a line to• wards which the curve under consideration is constantly approaching when tending to infinity. The word "asymptotos" in Greek means "non• coincident." But the fact of non-coincidence alone does not cover the idea of asymptotics and, etymologically, the term seems insufficient. However, it puts strong emphasis on the point that approximation does not turn into coincidence, and it is this salient feature of approximation not converting to coincidence that characterizes asymptotic phenomena in a wide sense of the word. In physics and other fields of science we constantly come across pro• cesses of asymptotic nature, for instance such as damping, orbiting, sta• bilization of a perturbated motion etc. Once the parameter of a process is understood to mean, as the occasion demands, not only time but any relevant variable, the domain of asymptotic phenomena gets noticeably extended. It can also include, for instance, the action of an explosion over great distances, the behavior of materials near boundaries, and the effects of small viscosity on fluid flow. A common property of such phe• nomena is the presence of specially separating subspaces, the vicinity of which is being investigated with respect to the phenomena under con-
215 216 ASYMPTOTOLOGY: IDEAS, METHODS, AND APPLICATIONS
sideration. In the above listed examples these are an infinite point, the material boundary, and non-viscous fluids. Let us now consider asymptotic phenomena in a wider sense. Gen• erally speaking, any object of investigation is not homogeneous. There are domains where quantities undergo sharp changes: break, fracture, vanish, or tend to infinity, etc. A transition which involves qualitative change (instead of only (small) quantitative changes) (a transition from quantity to quality) is understood as a special case (peculiarity) and it is only natural that such domains are specified as singular. These may be points, lines, surfaces and, in general, any variety of dimension m < n where n is the number of independent coordinates and parameters of the object under consideration. An asymptotic situation arises when the vicinities of singular varieties investigated are not fixed but decreas• ing ones. Phenomena. typical of such an approach are commonly called asymptotic. It seems that this definition has no claim on being rigorous for it includes the notions of specificity and singularity, which are rather ten• tative and preliminary. In fact, it is not yet clear what change in quantity counts as qualitative and what kind of behavior is specific. It would be premature to make haste in an attempt to get rid of the tentativeness if we try to grasp the essence of the asymptotic approach. The lack of definitiveness which persistently accompanies asymptotic thinking draws to it the attention of mathematicians of a. romantic stamp and turns away advocates of a more rigorous persuasion. That is the reason why the history of asymptotic mathematics was not monotonous (in both meanings of the word). After H. Poincare had introduced the notion of asymptotic series in 1886, new chapters, devoted to asymptotic expansions, appeared in university courses on mathematical analysis. The first half of our century, however, was not a time of real flowering for asymptotic methods. By the middle of the century the available material concerned with asymptotics did not get consolidated into monographs and even gradually disappeared from the majority of courses. Students graduating from universities were not acquainted with such things as the saddle-point method and the Stokes' phenomenon. In the last decades this state of affairs has changed considerably. The influence of asymptotic methods in applied mathematics has increased many times. The methods of perturbations, averaging, and bound• ary layers became the topics of monographs. Quasi-classical mechanics, diffraction theory, and shell theory developed, thanks to the asymptotic approach. The vitality and prospects of asymptotic methods become obvious from the fact that an active interaction between numerical and analytical methods is accomplished via asymptotics. Appendix D: Asymptotic Approaches: An Attempt at a Definition 217
This revival of asymptotic activity drew attention to the methods themselves, to those general features of asymptotics with may be studied irrespective of the subject of application. In this connection, it was found that the notion of asymptotic methods persistently defies a sufficiently rigorous definition. When N. de Bruijn, 1981 tried to answer the question "What is asymptotics?" he found nothing better than to call asymptotic estimates a section of analysis dealing with problems of the same type as those considered in his book (de Bruijn, 1981). Of course, enumeration with dots cannot be regarded as a satisfactory definition, especially as, on the one hand, asymptotic estimates pervade all branches of mathematics and, on the other, asymptotic methods are not restricted to estimates which have been formalized in any way. It seems that a real specification of what asymptotics is should be searched for not in some field of mathematics or science but in some methodological domain. But then one finds that there is a store of asymptotic methods, that they comprise a number of approaches, and that some are applicable rather to the arts than to science (Babic and Buldyrev, 1982). M. Kruskal, 1963 even introduced a special term "asymptotology" and defined it as an art of handling applied mathe• matical systems in limiting cases. Here it should be noted that he called for a formalization of the accumulated experience to convert the art of asymptotology to science. Thus, the difficulty seems to be the fact that formal definitions are too narrow while sufficiently wide definitions never meet the requirements of being truly scientific. It seems that there might be reasons for the fact that the existing concrete methodologies defied strict definitions within the framework of classical mathematics. Let us try to define asymptotic methods from a point of view, pri• marily using a criterion of adequacy of description of the real objects without giving in to the Procrustean bed of absolutely strict definitions. As a first approximation it would be most simple to term asymptotic those methods which are used for the study of asymptotic phenomena. But that is not a way to discover their content. The aim of the asymptotic approach is to simplify the object. This simplification is attained by decreasing the vicinity of the singularity un• der consideration. It is typical that the exactness of asymptotic expan• sions grows with localization. Exactness and simplicity are commonly regarded as antagonistic and complementary notions. When tending to simplicity we sacrifice exactness, and trying to achieve exactness we ex• pect no simplicity. Under localization, however, the antipodes converge, the contradiction is resolved or, in other words, eliminated, in a synthesis which is called asymptotics. 218 ASYMPTOTOLOGY: IDEAS, METHODS, AND APPLICATIONS
Thus the essence of asymptotic methods is that they accomplish a the synthesis of simplicity and exactness by means of localization: in the vicinity of a certain limiting state one finds a simplified solution of the problem which is the more exact the smaller the vicinity. As has been noted by Laplace, the more indispensable the asymptotic methods the more exact they are. In fact, a question arises where global methods fail to work, but, in any case, it is there, in the vicinity of singularities, where they are most efficient. Why do asymptotic methods put a claim on being more then a part of classical mathematics? What did it mean when K. Friedrichs (1955) said: "Asymptotic description is not only a convenient tool in the math• ematical analysis of nature, it has some more fundamental significance"? All this is worth thorough consideration. In practice, we use asymptotic methods not only for the solution of clearly formulated problems but also for the statement of problems and, in general, in the process of trying to understand the world. Although most everything is interrelated in nature the relations are not uniform and, due to this nonuniformity, it is possible to single out and study relatively isolated systems. But those systems themselves may be re• garded as singularities in the domain of universal interrelatedness, and their separation as localization in this domain. So the statement of an isolated problem looks like the localizing of a singularity and a corrected statement as a study of the vicinity of this singularity. Thus asymptotic methods go beyond mathematics. But in mathe• matics they occupy a special position since they find no definite place in any orthodox classification. Combining the simplicity of heuristic notions with exactness of analytical estimates, asymptotic methods are not confined to any kind of "golden mean." Their main difference from the methods of classical mathematics is that the level of their exactness competes with the size of the action domain. In a given domain the exactness of asymptotic expansions is limited. The ideal of classical mathematics is absolute exactness, and from this viewpoint any restriction of exactness is looked on as a deficiency. But would not absolutization be a withdrawal from life as human activity is not eternal and the truth is never complete? Will an inaccessible ideal turn out to be a drawback from the standpoint of the dynamics of life? And is the absolute exactness of classical mathematics is not tantamount to a limitation? It is evidently high time to look at asymptotic mathe• matics as a worthy competitor in the struggle for adequate quantitative descriptions of our dynamic reality. Appendix D: Asymptotic Approaches: An Attempt at a Definition 219 2. UNCERTAINTY-COMPLEMENTARITY- COMPATIBILITY As it has been mentioned above, asymptotic methods perform a syn• thesis of simplicity and exactness via localization. By fixing the size of a domain we restrict the possibilities both of simplification and specifi• cation. In other words, simplicity and exactness are connected through a complementarity relation while the domain size serves as a measure of uncertainty. Let as show that there is an uncertainty relation to be found for each pair of the three components of asymptotics and the third always plays the role of a regulator. Let us take an expansion of the function f (x) in an asymptotic se• quence { cpn (x)}
00 f(x) rv L ancpn(x), X -+ 0. (D.l) n=O
A partial sum of the series is designated by S N ( x) and the exactness of approximation (at a given N) is estimated by
LlN(x) = lf(x)- SN(x)j. (D.2) Simplicity is characterized here by the number N, and the locality by the length of interval for x. Now let us consider in pairs the interrelation of x, N, 6. basing our• selves on known properties of asymptotic expansions. At a fixed x the expansion initially converges, i.e., the exactness increases at the cost of simplicity. If we fix N, the exactness and the interval size begin to compete. The smaller the interval, the simpler the given value of 6. is reached. Let us illustrate these regularities using an example. We consider the integral exponential function
(D.3)
Integrating by parts we obtain the following asymptotic expansion at y -+ -00.
00
Ei(y) rv eY L(k- l)!y-k. (D.4) k=l Put
f(x) = -e-Y Ei(y), y = -x-1 . (D.5) 220 ASYMPTOTOLOGY: IDEAS, METHODS, AND APPLICATIONS
By calculating the partial sums (D.4) and the value (D.2) for different x we obtain a table D.1 where m is the value Nat which optimal exactness is achieved.
Table D.l.
x-t f(x) 6.m(x) m x-1 f(x) 6.m(x) m 3 0.262 0.034 3 7 0.127 4.0 w-4 7 4 0.207 0.011 4 8 0.112 1.4 10-4 9 5 0.170 3.7 10-3 5 9 0.101 2.9 10-5 9 6 0.145 1.2 10-3 6 10 0.092 1.8 w- 5 11
Thus at a given x the simplicity increases with the growth of N up to m(x). By fixing N one may observe an improvement of exactness with diminishing x. The given exactness .0. is attained further with an increase of N as the domain becomes larger, and, what is most interest• ing, is that beginning with some x it is not achieved at all. Limited exactness is an essential property of asymptotic mathematics, making it different it from classical mathematics. Only in the case of convergent series may absolute exactness be attained at a given x, but only by means of absolute complexity, i.e., at N ---+ oo. In the gen• eral case the uncertainty is of principal importance (Barantsev, 1982). Therefore, asymptotic mathematics may be related to classical mathe• matics in the same way as quantum mechanics to classical. To comprehend this analogy better, let us take the point of view of "system triads", a particular case of which is the asymptotic triad "exactness-locality-simplicity" (Barantsev, 1980; Barantsev, 1984). In general, a system triad expresses the unity of the rational, emotional, and intuitive sides of thinking, reflecting the analytical, qualitative, and substantial aspects of cognition. Complementarity, a trait inherent in the components of asymptotics, is common trait in all system triads. The Heisenberg uncertainty relation "reconciles" coordinate and mo• mentum, the corpuscle and wave aspects via the Planck constant, which relate to the frequency characteristics of the object. In other words, it combines the statics and dynamics from aspect of rhythm. Thus the structure of the triad systems in asymptotic mathematics and quantum mechanics differs only in orientation: a qualitative measure dominates in the first, and a substantial one, in the second. Partisans of the classical paradigm are always inclined to get rid of an "annoying uncertainty" through searching for hidden parameters. The Appendix D: Asymptotic Approaches: An Attempt at a Definition 221
principle of uncertainty, however, has withstood all such attempts to make things deterministic thus displaying its fundamental ontological basis. Nonetheless, discussions are still in progress. The last few years they have been centering on Bell's inequality for correlation functions of triple measurements (Spassky and Moskovsky, 1984). It follows from the inequality that a theory of hidden parameters reproducing all results of quantum mechanics has to be essentially non-local. Thus there arises an opposition "determinism-locality" the quests for the solution of which lead to the notion of wholeness. In the triad "elementness-relatedness-wholeness" (Barantsev, 1980) the last component expresses the substantial aspect of the system. This aspect, meaning the wholeness, ought to be differentiated from its inte• gral synthetic meaning. Closing is predominant in the synthesis while the substantial aspect is inevitably opened. Absolute closing as much as absolute openness are ruled out for they distort the notion of whole• ness as such. This new manifestation of the uncertainty principle means, among other things, that wholeness is incompatible with completeness (Bohm and Hiley, 1975), a tendency which has always been present in the strategy of scientific search (Bazhanov, 1983). Consequently, in order to find wholeness one must give up on com• pleteness. From what position and to what measure can one solve the paradox? Let us change the topic a bit, for the sake of sharpness and freshness, and not consider a concrete object, but turn to the genre of confession, the classical examples of which were given by St. Augustin, Rousseau, Leo Tolstoy. The ideal of perfection, purity, completeness or• ders us to attempt the utmost sincerity, leaving unheeded censorship, onlookers, human judgment. But if you make an attempt to eliminate this background, confession is doomed. It is the same as with theater where conventionality is indispensable as much as it is earlier in chil• dren's games (Gulyga, 1986). In our quest for completeness we, at a crucial moment, begin to withdraw from active life. This means that it is time to replace the static ideal for a dynamic one. The contradic• tion between completeness and wholeness is resolved via dynamics in the same fashion as the opposition exactness-simplicity is resolved via localization. Tending to perfection frequently leads us up a blind alley, the escape from which inevitably results in the denial of absolute certainty. The po• tential equality of the aspects of a system triad implies that any might be a blind alley, while for resolving any antithesis the third aspect is required. These regularities are easy to trace, for example, for such a popular triad as "duty-love-vocation." Another example is the coordi• nation of three kinds of activity: for myself, for society, for the sake of 222 ASYMPTOTOLOGY: IDEAS, METHODS, AND APPLICATIONS an idea; unfortunately, we are well aware of the extremities of egoism, social activity, fanaticism. Determinism can become an obstacle in the path to unification (Bondi, 1983; Kline, 1980; Popper, 1982). The principle of uncertainty keeps that path open. Absolute exactness is not an ideal but the same sin as absolute unexactness. Any theory having an invariable foundations can• not and must not suffer from a maximally precise formulation (Val'kov, 1975). As a formal instrument, asymptotic mathematics succeeds clas• sical mathematics. The term "uncertainty principle" reflects its meaning, but only par• tially. In the other hiepostasis the "complementarity principle" over• shadowed quite daringly the domain of quantum mechanics (Alexeev, 1978). Completing the triad definition we may propose a term for the substantial aspect as a "compatibility principle." In the paper (Strigachev, 1982) a conclusion is drawn about the fun• damentality and unity of three concepts of modern physics (comple• mentarity, correspondence, relativity) and their connection to conserva• tion laws is established. The uncertainty relation appears due to the conservation of the third value. In this content it should be stressed that consideration of the triad elements in pairs given a fixed measure is a contribution to the dyad paradigm. Simultaneous incorporation of all three components yields a symmetric form of the uncertainty• complementarity-compatibility principle. It is rather easy to make a formal generalization by multiplying three versions of the inequalities of the form 6x1 · 6xz ;:::: h3. As a result we obtain
(D.6) But one must still give meaning to the values involved. In case we restrict ourselves to physics, we come across a most curious fact when following Philbert's trichotomy (Philbert, 1974). Distinguishing the the• ory of elementary particles, relativity theory, and quantum mechanics we have respectively three limitation measures given by the constants e, c, h the known combination of which forms a dimensionless constant a= 1/137 (the fine structure constant) May this not be a structure that determines some universal scale H?
Notes 1 (Cauchy, 1868). Appendix E Some Web-Pages
"There are a number of commercial software packages with mathemat• ical visualization capabilities. Of these, perhaps the best known are The Three M's: Matlab, Maple, and Mathematica. The standard licenses for these programs are expensive, but there are also inexpensive student versions available, and many universities have site licenses. These are not primarily mathematical visualization programs. Maple and Math• ematica are symbolic manipulation programs, and Matlab is a numer• ical analysis program, but all three have very good graphic back-ends for displaying the results of their computations, making them excellent platforms for mathematical visualization. One minor drawback is that each of these programs has its own programming language that a user must learn in order to do anything nontrivial with them. But these are very high-level interpreted languages, and they are considerably easier to learn and to use than the standard compiled languages. An impor• tant point in their favor is that there are versions of each for Macintosh, Windows, and various flavors of UNIX, and software developed for any of these platforms is readily transportable to the others" (Palais, 1999).
On-line Mathematics Visualization Software
3D-Filmstrip Home Page, http:// rsp.math.brandeis.edu/ public_html Ken Brakke's Surface Evolver Home Page, http://www .susqu.edu /facstaff/b /brakke/evolver /evolver. html Geomview Home Page, http:/ /www.geom.umn.edujsoftware/download/geomview.html Grape Home Page, http:/ /www-sfb288.math. tu-berlin.de/ -konrad/ grape/ grape. html 223 224 ASYMPTOTOLOGY: IDEAS, METHODS, AND APPLICATIONS
Oorange Home Page, http:/ jwww-sfb288. math. tu-berlin.de/oorange SnapPea Home Page, http:/ jwww.geom.umn.edu/softwarejdownloadjsnappea.html Superficies FTP Site, ftp: / /topologia .geomet. uv.es/ pub/ montesin /Superficies_Folder / Surf Home Page, http:// www.mathematik.uni-mainz.de/ AlgebraischeGeometriejsurf / surf.shtml Mathworks (Mat lab) Home Page, http://www. mathworks.com/ products/ matla b/ Maple Home Page, http:/ /www.maplesoft.on.ca Wolfram Research (Mathematica) Home Page, http:/ jwww.wri.com/ References
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Igor V. Andrianov was born on April 19, 1948. Igor Andrianov is currently Professor of Higher Mathematics at the Pridneprovskaya State Academy of Civil Engineering and Architecture (PGASA), Dne• propetrovsk, Ukraine. Andrianov graduated from the Dnepropetrovsk State University (DGU) (Mechanical and Mathematical faculty) in 1966. Andrianov received his Ph.D. (Mathematics and Physics) degree from the DGU in 1975 and his D.Sc. (Mathematics and Physics) from the Moscow Electronic Machine-Building Institute in 1990. He was post• graduate student of DGU, Senior Scientific Researcher of DGU, since 1977 Associate Professor and since 1988 Full Professor of PGASA. An• drianov is the author or co-author of over two hundred fifty papers (over 70 in international journals) and six books. His research interests include Asymptotic Methods, Mechanics of Solids. He is a member of AMS, GAMM, SIAM, and EUROMECH. SOROS Professor (1996). Hobby: writing of popular scientific papers. Leonid I. Manevitch was born on April 2, 1938. Leonid Manevitch is currently Head of Theoretical Division in N.N. Semenov Institute for Chemical Physics of Russian Academy of Sciences and Professor of Physics at the Moscow Institute of Physics and Technology. Manevitch graduated from the Dnepropetrovsk State University (DGU) (Physical and Mathematical faculty) in 1959. Manevitch received his Ph.D. (Tech• nical Sciences) degree in 1961 and his D.Sc. (Technical Sciences) from the DGU in 1972. Manevitch is the author or co-author of over three hundred fifty papers (over 100 in international journals) and ten books. His research interests include Nonlinear Dynamics, Polymer Physics, Asymptotic Methods, Mechanics of Solids. He is a member of AMS, GAMM, and EUROMECH. SOROS Professor (1995-96, 1997, 1998, 2001). Hobby: philosophy, history, writing of popular scientific papers-.
239 240 ASYMPTOTOLOGY: IDEAS, METHODS, AND APPLICATIONS
Michiel Hazewinkel was born in Amsterdam, the Netherlands in 1943 and grew up in Pontianak, Indonesia and Emrnen, the Netherlands. He studied at the University of Amsterdam and obtained his Dr c!egree in 1960. He has been a full professor of mathematics since 1972, first at Erasmus University, Rotterdam. Currently he is at the CWI (Center for Mathematics and Computer Science), Amsterdam. He has frequently been a guest at other institutes, notably the Steklov Institute for Mathe• matics, Moscow (ZWO grant, 1969-1970) and Harvard University (1974, Fullbright-Hays grant). He is editor of the Encyclopaedia of Mathe• matics (13 volumes, 1988-2001, Kluwer Academic Publishers) aud the Handbook of Algebra (Elsevier, since 1995) and has authored well over a hundred scientific books and papers. Author Index
Abel, N.H., 11, 154 Bernoulli, D., 146 Ablowitz, M.J., 4-5 Bernoulli, Jakob, 139-140 Ackeret., J., 17il Bernoulli, Johann, 139, 143 Adams, J.C., 82, 134-135 Berry, M., 25, 39 Akhundov, M.D., 101 Bianci, L., 175 Alembert, J. d', 14:1-147, 151, 185 Birdzell, L.E., Jr., 101 Alexandrov, P.S., 158 Birkhoff, G., 103 Alexeev, I.S., 222 Blekhman, I.I., 38 Andrianov, f.V., 17, 38, 64, 8.5, 119, 126, Bogoliubov, A.N., 95, 178 178 Bogoliubov, N.~., 17,129, 133, 173-177 Andronov, A.A., 70 Bohler, W.K., 129, t78 Apollonius of Perga, 39 Bohm, D., 221 Appell, P., ~58, 165-167 Bohr, N., 103 Archimedes, 39, 168 Boltzmann, L., 30, 116, 126 Aristotle, 104-107 Bolyai, J., 156, 159 Arnol'd, V.i., 4-5, 72, 153, 168, 177, 204, Balzano, B., 154 206 Bondi, H., 222 Ashby, W .R.., 94 Borel, F.E.J E., 15-16, 25, 154, 165-167 Augustin, St., 221 Bosley, D.L., 35 Awrejcewicz, J., 38, 64, 119 Bourbaki, N., 11, 36, 129 Bourgat, J.F., 18-19 Babic, v .'vl., 217 Boussinesq, J., 167, 175 Baker, G.A., Jr., 25, 28, 168 Bracken, P., 25, 32 Bakhvalov, N.S., 17 Brahe, T., 116 Barantsev, H..G., 102, 215, 220-221 Brandt, S., 97 Barashenkm·, V.S., 1, 3 Braque, G., 96 Baravalle, H. von, 96 Brezinski, C., 178 Barenblatt, G.!., 11 Brillouin, L., 109--110 Barrow, I., 168 Bronshtein, M.P., 113 Basset, A.B., 125 Brown, H.., 90 Batchelor, G.K., 36 Bruijn, N.G. de, 41, 217 Bazhanov, V.A., 221 Bruning, J., 7 Bazhenov, L.B., 101 Bruno, A.D., 65-66, 68 Beletsky, V V., 83 Bubnov, !.G., 36 Bellman, R., 94-95 Buchhardt, G., 90 Bender, C.M., 62-64 Buldyrev, V.S., 217 Bensoussan, A., 17 Butuzov, V.F., 95 Berdnikov, V.A., 102 Berdyayev, I\' .A., 97 Cezanne, P., 95 Berkov, A.V, 108 Calladine, C.R., 119, 129 241 242 ASYMPTOTOLOGY: IDEAS, METHODS, AND APPLICATIONS
Caramazza, A., 104 Fermat, P., 168 Caratheodory, C., 141 Fermi, E., 183, 190 Carlini, F., 110 Filimonow, A.M., 22, 60 Carno, N.L.S., 149 Fleck, G., 95 Carrier, G.F., 129 Fleming, J.A., 172 Cartan, E.-J., 165-166, 174 Fock, V.A., 98-99, 112, 171 Catalan, E.C., 35 Fomenko, A.T., 97-98 Cauchy, A.-L., 122, 146, 154, 156, 168, 215 Fourier, J., 22-23, 116, 131, 150, 154, 157, Cavendish, H., 172 185, 194 Cesaro, E., 25 Frederick II, 140-141 Chebyshev, P.L., 161-163, 175 Frenkel, V.Ya., 31, 129 Chernin, A.D., 30-31 Frenkel, Ya.I., 195 Chirikov, B.V., 10 Fresnel, A.-J., 116, 186 Chladni, E.F.F., 121 Freud, S., 99 Christensen, R.M., 91, 93 Frey, D., 96 Churchland, P.S., 100 Friedrichs, K.O., 111, 218 Cizek, J ., 25, 32 Frobenius, G., 168 Clairaut, A.-C., 129, 131, 134, 140, 142-144 Clement, J., 104 Galerkin, B.G., 36 Cole, J.D., 129 Galileo, 4, 101, 104, 106, 113, 135 Condorcet, J ., 149, 151 Galle, I.G., 135 Copernicus, N., 82, 117 Gamow, G., 30, 112 Cosserat, E., 121, 165 Gans, R., 110 Cosserat, F., 121 Gardner, C.S., 191 Cramer, G., 140 Gauss, C.F., 97, 129, 155-157 Gelfand, I.M., 93, I 91 Dahmen, H.D., 97 Gell-Mann, M., 24 Dana, C.A., 182 Gennes, P.-G. de, 112 Danilov, V.G., 89-90 Germain, S., 121 Darboux, G., 158, 165, 167, 175 Gilewicz, J ., 164, 166 Darwin, Ch.R., 126, 153 Ginzburg, V.L., 82, 95, 138 Darwin, G., 153 Gol'denveizer, A.L., 119, 123, 125 Delane, C., 129 Goldstein, S., 170 Denjoy, A., 174 Gordon, J.E., 83 Descartes, R., 8, 66, 68, 115, 207 Gordon, W., 192, 195 Dewar, M.J.S., 95 Grave, D.A., 176 Diderot, D., 146-147 Graves-Morris, P., 25, 28, 168 Dini, U., 175 Grebennikov, E.A., 129, 134 Dirac, P.A.M., 31 Gredeskul S.A., 88 Dirichlet, P.G., 26 Green, B., 104 Dostoyevsky, F.M., 97 Green, G., 110, 186 Dvortsina, N.S., 112 Green, J., 4-5 Eddington, A., 144 Greene, J.M., 191 Einstein, A., 2-3, 90-92, 97, 104, 106, 116, Grimsley, R., 178 119, 127, 137, 144, 160, 177-178, 181 Gromov, M., 179 Elishakoff, I., 117, 179, 182 Grosberg, A.Yu., 88 Eri, J.B., 134 Gubanova, !.I., 54 Escher, M.C., 98 Guckenheimer, J., 34, 179, 182 Euclid, 101, 106, 156, 186 Gulyga, A.V., 221 Euler, L., 25, 78, 129-130, 133, 138-143, Gustafson, K., 110 146, 148-149, 152, 155-156 Euler, U.S. von, 141 Hacken, H., 10 165-166, 175 Euler-Chelpin, H.K.A.S. von, 141 Hadamard, J.-S., Halley, E., 142-143 Faddeev, L.D., 192 Hamilton, W., 107, 149 Fatou, P., 133 Hamming, R.W., 179, 181 Fedoryuk, M.V., 42 Hawking, S.W., 3, 9, 35, 81 Feodos'ev, V.I., 36 Heaviside, 0., 23, 173 Author Index 243
Heisenberg, W.K., 220 Landau, L.D., 44, 112, 127 Hermite, Ch., 160, 165-166 Langer, R.E., 110 Hilbert, D., 97 Laplace, P.-8., 8, 17, 34, 129, 131, 138-139, Hiley, B., 221 144, 146, 149-154, 173, 185, 218 Hill, J., 129 Lavoisier, A.-1., 149, 151 Hinch, E.J., 110 Le Verrier, U.J.J., 82, 129, 134-135 Hooke, R., 116-117 Lebesgue, H.-L., 165-166, 175 Hopfinger, E.J., 36 Leibniz, G.W., 68, 115, 148, 156, 168 Horn, J., 110 Lenat, D.B., 100 Hospital, G.F. de 1', 140, 143 Lensel, A., 139 Hunt, G.W., 10 Lesk, A.M., 94 Lesnichaya, V.A., 17, 85,126 Ivanenko, D.D., 112 Levitan, B.M., 191 Ivanitskii, G.R., 94 Lifshitz, I.M., 88, 112 Lin, S.S., 41 Jacobi, K.G.J., 110, 129, 131, 168 Lindstedt, A., 16, 159 Jeans, J., 30 Lions, J.-L., 17 Jeffreys, H., 110 Liouville, J., 110 Jones, W.B., 25 Lobachevsky, N.I., 156, 161 Jordan, M.-E.-C., 167 Lorentz, H.A., 160, 178 Jost, R., 63 Lorenz, L.V., 171 Kalachev, L.V., 95 Love, A.E.N., 83, 119, 122-125 Kandinsky, W.W., 96 Lowell, P., 83, 135 Kantorovitch, L.V., 36 Lozanov, G., 97 Kaplunov, J.D., 119, 123 Luke, St., 102 Katz, M., 16 Lyapunov, A.M., 161-163 Kepler, J., 116, 184 Khaykin, S.E., 133 Mach, E., 101, 115, 126 Khesin, B.A., 5 Maclaurin, C., 25, 48-50, 60 Khokhlov, A.R., 88 Malevich, K.S., 96 Kirchhoff, G.R., 83, 122-125 Malinetskii, G.G., 100 Kirchraber, U ., 129 Mandelshtam, L.I., 4, 54, 127, 129, 133 Kirzhnits, D.A., 31 Manevitch, L.I., 17, 38, 60, 64, 68, 84-85, Klein, C.F., 166 126, 183 Kleiner, A., 90 Marchenko, V.A., 191 Kline, M., 106, 116, 135, 222 Marchuk, G.I., 86 Koblik, S.G., 84 Maslov, V.P., 89-90 Kobzarev, I.Yu., 108 Maupertuis, P.-L.M. de, 140, 143 Kolmogorov, A.N., 153 Maurin, L.N., 198, 200 Kontorova, T.A., 195 Maxwell, J .C., 92, 99, 113, 133, 186 Koptsik, V.A., 95 McClosky, M., 104 Korteweg, D.J ., 26, 188 Medvedev, F.A., 178 Kossovich, L.Ya., 119, 123 Meshkov, I.N., 10 Kovalevskaya, S.V., 16 Meyer, R.E., 110 Kraft, V., 139 Migdal, A.B., 1-2, 29-30, 36, 87, 95 Kramers, H.A., 109-110 Mikhlin, Yu.V., 60 Kruskal, M.D., 4-5, 190, 217 Milne, A.A., 42 Krylov, A.N., 89, 149 Milton, K.A., 62-64 Krylov, N.M., 129, 133, 173-176 Minkowski, G., 178 Kurchanov, P.F., 22, 60 Mitropolsky, Yu.A., 17 Kurdyumov, S.P., 100 Mittag-Leffler, M.G., 16 Kuznetsov, B.G., 97, 135 Miura, R.M., 191 Moiseev, N.N., 34, 89, 94, 104, 116, 163 Lagrange, J.-L., 78, 129-131, 133, 139-140, Monge, G., 149 146, 148-152, 162 Morgan, S.P., 181 Lalande, J.-J., 143 Morse, H.M., 202 Lamb, W., 125 Moser, J .K., 153 Lambert, F., 26 Moskovsky, A.V., 221 244 ASYMPTOTOLOGY: IDEAS, METHODS, AND APPLICATIONS
Murray, J.D., 93 Ptolemy, C., 117 Musette, M., 26 Myasnikov, V.P., 89-90 Rajaraman, R., 16 ll!yshkis, A.D., 22, 38, 60 Raushenbakh, B.V., 96 Rayleigh (Strutt), J.W., 4, 30, 110, 124-125 Nabokov, V.V., 126, 180 Reed, M., 16, 63 Nalimov, V.V., 180 Renoir, A., 95 Napoleon I, 121, 150, 153-154 Reuss, A., 18-19, 93 Navier, C.L.M.H., 116, 122-123 Reynolds, 0., 170-171 Nerubaylo, B.V., 85 Riemann, B., 146 Nesterenko, E.M., 178 Rosenberg, N., 101 Neumann, J. von, 127 Roseveare, N.T., 109, 129 Nevanlinna, R., 101, 127 Rousseau, J.-J., 221 Newcomb, S., 16 Rozenfel'd, B.I., 93 Newton, 1., 46, 62, 65, 68, 82, 101, 104-105, Rozental', 1.1., 3, 9, 35 107-108, 113, 116, 122, 133-134, 138, Russell, J.S., 187, 190 140, 142, 144, 155-156, 168, 178, 186 Ryabov, Yu.A., 129, 134 Nolde, E.V., 119, 123 Novozhilov, I.V., 41, 97 Sagdeev, R.Z., 24 Saint-Venant, A.I.C.B., 122 O'Malley, R.E., Jr., 129, 170 Sakharov, A.D., 177 Obraztsov, I.F., 85 Schrodinger, E., 21, 110, 192 Ohm, G.S., 116-117 Schreyder, Yu.A., 98 Okun', L.B., 114 Segel, L.A., 41, 62, 169 Olver, F.W.J., 110 Segur, H., 4-5 Selye, H., 97 Pade, H., 25-26, 28-29, 31-32, 38, 47·-48, Senechal, M., 95 63-64, 86, 91, 93, 164-168 Shabat, A.B., 192 Painleve, P., 160, 165-166, 175 Shafarevich, I.R., 137 Palais, R.S., 223 Shafranovsky, I. I., 95-96 Panasenko, G.P., 17 Shakespeare, W., 138, 149 Panovko, Ya.G., 38, 54 Shamrovsky, A.D., 207 Papaleksi, N.D., 129, 133 Shanks, D., 25 Papanicolaou, G., 17 Shibaev, V.P., 112 Pascal, B., 168 Shibanov, A.S., 16, 159-160, 163, 178 Pasta, J.R., 190 Shifrin, M.A., 93 Pastur, L.A., 88 Shirkov, D.V., 177 Pauli, W., 112 Shubnikov, A.V., 95 Pavlenko, A.V., 84 Siegel, C.L., 153 Pedoe, D., 95 Simmons, L.M., Jr., 62-64 Peierls, R.E., 114, 116 Simon, B., 16, 63 Pekosh, N.R., 143 Skovoroda, G.S., 81 Philbert, B., 222 Sonin, A.S., 112 Piazzi, G., 156 Spassky, B.!., 221 Picard, C.-E., 16.5-167, 175 Spirko, V., 25, 32 Picasso, P., 96 Steklov, V.A., 162-163 Pilipchuk, V.N., 60 Stewart, 1., 111, 197, 204-206 Pinsky, S.S., 62-64 Stieltjes, T., 11, 154, 166 Planck, M., 29-30, 81, 112, 169, 220 Stokes, G.G., 116, 216 Poincare, J.H., 4, 11-12, 16,27-28, 72, 103, Strigachev, A., 222 116, 127, 129, 137, 153-154, 157-163, 165, 167, 178, 216 Takehisa, A., 110 Poincare, L., 157, 165 Tamm, I.E., 177 Poisson, S.-D., 122-123, 153 Tannery, J., 165 Pomeranchuk, I.Ya., 95 Tannery, P., 165 Pontryagin, L.S., 71 Taylor, B., 168, 197 Popper, K.R., 222 Thompson, J.M.T., 10 Poston, T., 111, 197, 204-·206 Thoreau, H.D., 111 Prandtl, 1., 125, 169-171 Thron, W.J., 25 Author Index 245
Timofeev-Resovsky, N.V., 81 Vries, G. de, 26, 188 Timoshenko, S.P., 121 Tisserand, F.-F., 158, 160 Wall, H.S., 164 Titov, P.A., 89 Wataghin, G.V., 31 Tiulina, I.A., 178 Weber, W., 156 Tolstoy, L.N., 221 Weierstrass, K.T.W., 16, 159-160 Tombo, K., 135 Weinberg, S., 3, 32 Tomilova, A.E., 110, 129 Weniger, E.J., 25, 32 Truesdell, C.A., 126, 138 Wentzel, G., 109-110 Tsieu, H.S., 39 Weyl, H., 3, 95-96, 161 Turbiner, A.V., 34 Whitney, H., 73 Tyapkin, A.A., 16, 159-160, 163, 178 Wien, W., 29-30 Tzykalo, A.L., 129, 178 Wiener, N., 173, 182 Wigner, E., 95 Ulam, S.M., 190, 196 Wilcox, D.C., 95 Urbanskii, V.M., 95, 178 Wilson, K.G., 32, 34-35, 95 Usikov, D.A., 24 Wlassow, W.S., 119, 124 Vainstein, L.A., 99 Yaffe, L.G., 63 Vakakis, A.F., 60 Yaglom, l.M., 98 Val'kov, K.l., 222 Yudovich, V.I., 24 Van den Berg, I., 26 Yukalov, V.I., 32 Vander Pol, B., 129, 131-133, 172-175, 178 Yukalova, E.P., 32 Van Dyke, M., 13, 25, 43, 129, 170 Yushkevich, A.P., 129, 178 Van Loan, Ch., 117 Van Wijngaarden, L., 36 Zabusky, N.J., 4, 190 Vasil'eva, A.B., 95 Zakharov, V.E., 192 Vedenov, A.A., 89, 98, 111 Zaslavskiy, G.M., 24 Verhulst, F., 129-130 Zeldovich, Ya.B., 11 Verne, J., 41 Zel'manov, A.L., 113 Voigt, W., 18-19, 93 Zevin, A.A., 60 Voltaire, 143, 147 Zeytounian, R.Kh., 95 Volynskii, L.N., 9.5 Zhilin, P.A., 121 Vorontsov-Vel'yaminov, V.A., 129, 178 Zhizhin, E.D., 108 Zhukovskii, N.E., 22, 111 Topic Index
analogy, 3, 18, 23, 26, 38, 42, 50, 52, 54, 67, asymptotic splitting, 54 84, 89, 98, 110-111, 171, 186, 190, asymptotics, 7, 24, 27, 31, 33-34, 38, 41-42, 192, 198, 220 47, 58-59, 86, 88-89, 94-95, 97-98, approximant Pade, 25-26, 29, 31-32, 47-48, 102, 107-108, 110-111, 114, 116, 126, 63-64, 164, 166, 168 215-217, 219 approximation, 9-10, 14, 20, 23-24, 26-28, Aristotle, 107 32-33, 35, 37, 47, 50, 52, 56, 58-60, guessed, 123 64-65, 85, 87, 90-91, 99, 101, intermediate, 11 106-107, 109, 114, 116, 130, 134, 139, limit, 27, 29-30, 32 162-163, 169, 215, 219 motion, 106-107 asymptotic, 14, 42, 95 natural, 41 accuracy, 37 nonlinear, 64 first asymmetric, 95 regular, 9, 47, 49 difference, 59, 61 singular, 47, 49, 123, 125 finite-order, 31 small times, 108 first, 83, 96, 108-109, 123, 130, 135, 139, asymptotology, 4, 217 142, 162, 175, 217 atmosphere, 95 geometrical optics, 109 averaging, 16-18, 33-34, 82-83, 90, 94, 129, higher, 142, 174-175 131, 133, 149, 151, 173, 175, 177, 216 long wavelength, 190 Reuss approach, 18 method, 81 Voigt approach, 18 Pade, 25, 28-29, 38, 63-64, 86, 91, 93, 168 boundary layer, 2, 9-11, 35, 54, 86, 98, 110, Pade-like, 64 116, 123, 125, 169-171, 216 parabolic, 67 internal, 10 piecewise-linear, 96 boundary-value problem, 41 quasi-classical, 111 Chernobyl disaster, 89 second, 109 complementarity principle, 42 "shallow water", 188 composite materials, 18, 20, 90 strong coupling, 63 computer, 10, 34-36, 86, 90, 94, 100, 117, weak coupling, 62 143, 179-182, 190 zero, 38 conditions area of applicability, 83, 107, 115, 126 boundary, 11, 124, 185 art, 4, 26, 95, 97, 112, 114, 126-127, 147, Cauchy-Riemann, 146 169, 172,217 d'Alembert-Euler, 146 classical, 97 external, 104 fine, 95-97 extremum, 68, 74, 80 modern, 96-97 initial, 11, 51-52, 54-55, 185, 191 asymptotic equalities, 14 invariance, 207 asymptotic reduction of dimension, 57 continualization, 123 247 248 ASYMPTOTOLOGY: IDEAS, METHODS, AND APPLICATIONS continuation nonlinear, 80, 193 analytic, 25, 38 parabolic, 99 meromorphic, 25, 38 perturbated, 47 continued fraction, 25, 140, 166-167 quadratic, 54 continuous limit, 19, 21-22 quasi-linear, 163 convergence, 15, 31, 146, 154, 159, 163, Schriidinger nonlinear, 192 166-167 sine-Gordon, 192-193, 195 local, 31 soliton, 193 numerical, 31 structural-orthotropic theory, 85 correspondence, 222 transcendent, 74-75 asymptotic, 104-105, 107, 111 Vander Pol, 132, 173-174 decomposition, 94, 99, 181 wave, 109 approximate, 56 equations asymptotic, 56, 125 Kirchhoff-Love, 123 deformation of a function, 205 Maxwell, 99 denominator small, 152 transport, 109 Descartes folium equation, 207 estimation of errors, 23-24, 41-42, 47, 180 expansion, 23-25, 29, 33, 43, 46-47, 60, effect 62-65, 82, 86, 89, 109, 115, 123 edge, 83, 98, 125 asymptotic, 7, 33, 35, 43, 45, 62, 123, 159, nonlinear, 96 171, 175, 216-219 electrodynamics, 2, 113, 156, 186, 196 Fourier, 23 quantum, 82 limit, 28, 32 engineering, 9, 37, 73, 84, 86, 89, 93, 125, power, 68, 75, 77, 80, 198-201, 204, 207 165, 172-173 singular, 46 equation, 46-47, 49, 57, 82-84, 89-91, 94, strong coupling, 32 106, 112, 114, 120-121, 123-127, Taylor, 197 130-132 uniformly suitable, 46 algebraic, 8, 48 weak coupling, 32 4th order, 48, 80 5th order, 62-64 biquadratic, 46 fold catastrophe, 76 cubic, 65, 67-68, 79, 207 formula linear, 57 Newton-Leibniz, 168 nonlinear, 65, 68-69, 79, 206 Planck, 30 quadratic, 48-52, 79 Rayleigh-Jeans, 30 characteristic, 52, 54-55 Wien, 29-30 d' Alembert, 146, 185 function, 11-12, 14, 16, 18, 21, 25, 28, 31, differential, 8, 11, 17-18, 20,22-23,27,48, 42-43, 45, 54-55, 59, 61, 131, 219 51, 54, 78, 83, 90, 93, 114, 130-131, 1-defined, 199-200 134, 152, 160, 162, 166, 175, 184 2-defined, 199 2nd order, 19-20, 26, 51, 54-61 3-defined, 20 l 3rd order, 26 approximating, 28, 36 4th order, 58 Borel, 15-16 integrable, 111 comparison, 43 linear, 52, 109 correlation, 221 nonlinear, 65, 78, 109, 175 cubic, 79 partial, 146, 158 deformed, 205-206 dynamical, 159 elementary, 114 eikonal, 109 exponential, 15, 219 equilibrium, 75 gauge, 42, 44 Fourier, 185, 194 Green, 186 functional, 33 implicit, 65 heat conductivity, 92 linear, 184, 201 Kortcweg-de Vries, 26, 188, 190, 192 nonlinear, 201 Laplace, 185 periodic, 22, 130-131 linear, 185, 206 polynomial, 68, 75 motion, 107, 124, 146 quartic, 79 Topic Index 249
rational, 25-26 method structurally unstable, 78 abstraction, 136 transcendent, 75, 80, 197 analytical, 17, 35-36, 138, 216 trigonometric, 22 approximate, 24 asymptotic, 2, 4, 7-9, 11, 22, 24-25, 28, group, 32-33, 66-67 34-38, 41, 46, 61, 81-85, 88-90, continuous, 33 93-96, 137, 159-160, 171, 176, discrete, 33 215-219 renormalization, 32-35, 94, 177 boundary element, 36 engineering, 3 7 homogenization, 16-18, 85, 88, 94, 125, 181 finite element, 36-37 Reuss approach, 93 generalized summation, 38 Voigt approach, 93 Kirchhoff-Gehring, 124 Krylov-Bogoliubov, 176 idealization, 1, 9, 54, 101, 106, 111, 121, least squares, 155 127, 133, 135 Lindstedt-Poincare, 159 iteration, 44 mathematical, 93, 98 jet, 197-202, 204-206 multiple scales, 85, 181 Newton, 62 law, 81, 90, 96, 101 numerical, 17, 24, 28, 34-36, 179, 216 conservation, 1, 184, 191, 222 perturbations, 82 energy, 107 saddle-point, 216 Fourier, 116, 194 simplification, 81, 94 Hooke, 116-117 structural orthotropy, 125 motion, 17, 105, 184 summation, 16, 25 Newton variational, 24, 28, 36 cooling, 116 WKBJ, 109, 111 gravitation, 82, 133-134, 142 second, 107 Newton polyhedron, 66 Ohm, 116-117 nonuniformity, 45, 125-126, 218 laws Descartes, 115 Fresnel, 116 ocean, 34, 95 linearization, 22, 116-117, 175, 185 optics, 156, 183, 185 local, 23 Fresnel, 186 nonlocal, 23, 26 geometrical, 99, 109-111 nonlinear, 193 matching, 27-32, 93 wave, 109, 111, 113 asymptotics, 115 oscillations, 20, 22, 56-60, 84, 107, 178, 192 smooth, 28 amplitude, 22, 38 mechanics, 69, 73, 126, 130, 138, 140, 147, cylindrical shell, 124 149, 161-163, 169 flexural, 121 analytical, 130, 149 frequency, 38 applied, 167, 169 harmonic, 23, 51, 58, 149, 183, 185 Aristotle, 104-107 high-frequency, 61 celestial, 8, 12, 16, 82, 130, 137, 145, 149, long-wave, 59 151, 154-155, 158-160, 184 longitudinal, 19, 21-22, 60 classical, 104-105, 107-110, 113-114, 192 nonlinear, 133 continuum, 18, 122 normal, 22-23 deformable solids, 2, 35 pendulum, 22 fluid, 2, 25, 116, 125, 169-171 relaxational, 132 Hamiltonian systems, 107 saw-tooth mode, 22 nonlinear, 18, 174 self-excited, 173 quantum, 103, 109-111, 113, 186, 220-222 self-sustained, 186 quasi-classical, 216 short-wave, 20, 61 statistical, 113, 177 slow, 20 technical, 120 small, 186 theoretical, 149, 167 smooth mode, 59 250 ASYMPTOTOLOGY: IDEAS, METHODS, AND APPLICATIONS oscillator anharmonic, 31 eigenvalue, 34 Einstein, 92 padeon, 26 exactly solvable, 192 parameter, 2, 7, 9, 18, 23-25, 28, 31, 33, 38, extremal, 68-69, 72, 76, 80 46, 49, 57, 63-64 FPU, 190, 194 artificial, 38 geometrical, 96 asymptotic, 81 heat conduction, 10 control, 72, 76, 78-79, 203-206 homogeneous, 43 coupling, 56 instability, 95 dimensionless, 41 inverse, 82 external, 94 inverse scattering, 192 extreme, 34 isoperimetric, 141, 148 hidden, 220 linear, 191 internal, 94 motion stability, 162 large, 38, 41, 81, 86, 88, 90, 104, 181 nonlinear, 11, 80, 133, 184, 187, 191 perturbation, 45 optimal design, 86 small, 8, 14, 23-24, 33, 37-38, 41, 46-48, perturbed, 43, 52 62-63, 65, 69, 81, 83, 86, 88, 104, 112, physical, 2-3, 8, 16, 31, 33, 62, 81 115, 123-124, 126, 130-131, 139, 159, quantum-mechanical, 31 163, 181, 211 rational interpolation, 168 perturbation, 8, 11, 17, 24-25, 31, 33, 43, simplified, 38 45, 63-64, 83, 89, 112, 130, 133-134, singular, 50, 54 138-139, 145, 151, 156, 160, 205, 216 technical, 89, 120 large, 47 three-body, 8, 134, 159, 184 local, 17 two-body, 8, 184 periodic, 17 psychology, 93, 96,100-101,104-105,182 regular, ~' 49 singular, 43, 50, 106, 170 radius of convergence, 62-64 small, 46, 69, 88, 203, 205-206 reduction of dimension, 8-9, 19 theory, 8, 31-33, 38, 63, 82, 109, 138-139, asymptotic, 56-57 142, 177 renormalization, 32-33 quantum-theoretic, 63 quasi-linear, 26 scale, 33-35 phenomenology, 31, 104, 119-123, 125-127 separation, 16, 85 plate, 38, 86, 119-122, 170 asymptotic, 54 nonuniform, 125 sequence asymptotic, 43, 45-46 perforated, 85, 87 series, 24, 26-27, 33, 47, 62, 64, 162, 198, polymer, 86-88, 194-195 209, 219 popularization, 2-3, 160, 17 4 asymptotic, ll, 13-14, 42, 82, 137, 154, principle 159, 216 asymptotic simplification, 89 addition, 44 compatibility, 222 multiplication, 45 complementarity, 27, 42, 101, 222 subtraction, 44 continuity, 115 convergent, 11-13, 16, 24, 137, 166, 220 correspondence, 103 Dirichlet, 26 d'Alembert, 146 divergent, 11, 13-14, 16, 25, 154, 166 economy of thought, 101 Fourier, 22, 131, 186 first, 119-120, 122-124, 126 infinite, 14 heuristic, 100 Maclaurin, 25, 48-50, 60 least action, 148 perturbation theory, 32, 38, 63, 139 superposition, 11, 185-187 power, 25, 68 uncertainty, 221-222 semi-convergent, 154 problem, 41, 43, 45 Taylor, 159, 168 biological, 93 theory, 140, 146 boundary-value, 17, 26, 28, 36, 41 shell, 37, 83-84, 86, 119, 121-125, 127 complex, 2, 32, 35, 100 cylindrical, 124 dimensionality, 95 homogeneous orthotropic, 84 direct, 82, 86 nonuniform, 126 Topic Index 251
perforated, 85 characteristic, 84 reinforced, 85 equation, 46 ribbed, 85, 125 exact, 1 shallow, 38 internal, 185, 191 spherical, 10 latent, 23 stiffened, 84 spatial, 1 structurally-orthotropic, 38 system thin, 84, 121 1/2 degree of freedom, 52-53 vibrating, 124 analysis, 94 skin-effect, 2, 10 degenerate, 53-54 soliton, 4, 23, 26-27, 183, 187-195 disordered, 88 envelope, 192 limit, 2 equation, 193 living, 93 multi-dimensional, 196 two degrees of freedom, 56 solution, 9-10, 16, 20-24, 26-28, 31-33, 35-38, 42-43, 46-47, 49-52, 54-55, theory, 22 61, 63, 65, 81-82, 85-86, 90, 94-95, Brownian motion, 90 98, 100-101, 107-108, 112, 124, 126, catastrophe, 68-69, 71-73, 76, 80, 198, 130-131, 138 200, 202, 204-206 accurate, 31 diffraction, 98, 216 analytical, 1, 184 disordered systems, 88 approximate, 20, 33, 100, 124-125, 160, elasticity, 83, 119, 121-124, 165, 185 174-175 elementary particles, 222 everything, 113 asymptotic, 11, 23, 41, 48, 90 field, 32 asymptotic representation, 89 nonlocal, 31 classical, 108 quantum, 16, 113, 177 closed, 114 unified, 111 direct problem of the perturbation theory, general biological, 94 82 gravitation exact, 8, 11, 24, 26, 30-33, 47--50, 100, nonrelativistic, 113 106, 112, 114, 130, 132 quantum, 113 fast, 89 group, 186 fast changing, 126 interpolation, 175 general, 110, 191-192 Lyapunov-Poincan\, 163 homogenized, 85 molecular, 122-123 inverse problem of the perturbation nonlinear, 11 theory, 82 number, 140-141, 156 limit, 38 oscillation, 4 localized, 22, 25 physical, 1-2, 22, 104, 111-112 nonlinear, 26 plate, 83, 119, 121-122 numerical, 18-19, 31, 114 potential, 156 partial, 11 relativity, 108, 222 particular, 23 general, 9, 108, 113 periodic, 159, 163 special, 107, 113, 160, 178 power, 43 shell, 37, 83-84, 119, 121-127, 216 saw-tooth, 60-61 sound, 4 self-similar, 11 strong interactions, 82 simplified, 218 structural-orthotropic, 85 singular, 65-66 surfaces, 156 trial, 148 systems, 94 unique, 43 degeneration, 54 strong coupling, 56 thin films, 122 symbol, 97 tides, 153 Landau, 44 turbulence, 171 symmetry, 1-2, 7-8, 20, 33, 67, 77-78, vibration, 172 95-96, 98, 181, 184 thinking, 102 breaking, 1 asymptotic, 102, 114, 157, 216 252 ASYMPTOTOLOGY: IDEAS, METHODS, AND APPLICATIONS
emotional, 220 uniformity, 46 in1 uitive, 2:20 mathematicaL 186 variable contraction) SO physical, 114, 126 fast, 16 rational, 220 local, 17 scientific, 99 slow, 16 transformation spatial, 22 affine, 19 stretching, 49, 54~55 coordinate, 199 Euler, 25 wave, 30, 34, llO, 160, 172, 187, 189--190, Fourier, 157 191, 220 Pacle, 26, 91, 164, lti8 amplitude, 99 scaling, '207 gravitational, 95, 109 triad, 222 harmonic, 183~ 185, 187. 189, 194 asymptotic, 220 light, 109 weak coupling, 56 system, 220~221 Young modulus, 58