Appendix A Linear and Nonlinear Mathematical Physics: from Harmonic Waves to Solitons

Transcription

1 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 mathematicians. 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 harmonic 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 nonlinear mathematical physics, as it is called in the title of this section, as: 183

2 184 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 beginning 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 naturally 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 mechanics has properties, connected with of space-time symmetry, which are called conservation laws. Thus, the homogeneity of time causes conservation of full energy, the homogeneity of space-conservation of momentum, the isotropy of space (equivalence of all directions )-conservation of angular momentum. Every conservation law provides a certain relation between the sought-for functions which can be used for decreasing the dimensionality of the problem. In the two-body problem the conservation 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 theoretical 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.

3 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 transfer, and optics. Linearization turned out to be a very efficient universal procedure. Its efficiency has both mathematical and physical foundations. 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 consequence, new relations between variables. It is possible to find such a change of variables that splits the system of linear equations into independent 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 "collective" 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 oscillation 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 independent (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, electrostatics, and so on) the Laplace equation appears frequently in one or another form. In linear dynamics (acoustics, theory of elasticity, optics) the d' Alembert equation is the fundamental one; in relaxation processes (heat conductivity, diffusion) the Fourier equation plays such a role. These three equations constitute the basis of linear mathematical

4 186 ASYMPTOTOLOGY: IDEAS, METHODS, AND APPLICATIONS physics. This strongly effected the formation of mathematical thinking and development of mathematical notions (spectra of natural frequencies, 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 physicists simultaneously, created mathematical physics. While this takes place, the problems of theoretical physics constitute the most important 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 crucial 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 become 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 example, 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 electromagnetic 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 (stationary) 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

5 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 linear 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 formation 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 mathematical physics" has appeared in our days only. The point is that up to recent time there were no central notions in nonlinear problems similar 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 diminuation of speed." The language of this paper, unusual for us to hear, preserves 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 profile 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 dispersion). This absence of interaction as well as the dependence of wave speed on its amplitude is also characteristic of linear waves. It is possible 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 difference of speeds for harmonic waves of different length (dispersion).

6 188 ASYMPTOTOLOGY: IDEAS, METHODS, AND APPLICATIONS u /r D Fzgure kl. Spreading of wave packet due to dispersion, 11 is displacement of particles, x is spatial coordinate, t is time. This effect also manifests itself even in full absence of dissipation. Gravitational 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 approximation 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 nonlinearity). 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

7 Appendix A: Linear and Nonlinear Mathematical Physics X Figure A.2. Conservation of the profile of a solitary wave due to mutual compensation 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 hydrodynamics 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-

8 190 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 thermalization 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 something 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 onedimensional 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

9 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 presence 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 dimensionality 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.

10 192 ASYMPTOTOLOGY: IDEAS, METHODS, AND APPLICATIONS u 0 X Figure A.4. The envelope soliton describes the spatial localization of particles' oscillations 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 focus of interaction of, seemingly, distant fields of mathematics. This happened 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 nonlinearity 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 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-

11 Appendix A: Linear and Nonlinear Mathematical Physics 193 X Figure A.5. of a wave. Kink, soliton-like solution of the sine-gordon equation. Instant profile merous applications in many fields of Solid State Physics, Nonlinear Optics, and Particle Physics. The simplest soliton-like solutions describing 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 soliton equations is still increasing and their solutions lead to new physical ideas, connected with the particle-like properties of solitons. The universality 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-

12 194 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 prolongation 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 energy 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 velocity 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 relating 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 counteract 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 destruction 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 elasticity, 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 essentially 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-

13 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 intramolecular 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., localized along a line) structural defects in crystals. It models the simplest 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 distances. 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 unexpected 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 interactions 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

14 196 ASYMPTOTOLOGY IDEAS, METHODS, AND APPLICATIONS construction of a unified theory of fundamental interactions turn out to be essentially nonlinear, contrary to the classical electromagnetic theory A hypothesis that elementary particles are multi-dimensional solitons 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.

15 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 (x1, x2) the first jet is of the form: l aj I aj I J J = J(o,o) +-a x1 +-a x2 X1 0 X2 0 (B.l) 197

16 198 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 -l f 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 replacement of a function with certain of its jets so as to preserve a correct description 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 qualitative 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

17 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 expansion 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 singularities 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 transformation 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 = x 2, 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 equations for determination of the transformations T and r- 1. Their solutions 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 consideration. 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 concerning the local behavior of the function by studying a much more simple

18 200 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 expansion all such transformations can be obtained in terms of power series expansions in the independent variables, containing the linear terms (although the expansions of the functions can include every power). Moreover, 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 transformation 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 transformation we have to solve the equation r- 1 p f(x) = f(x). Because r-1 j 1j(x) = j 1j(T-1x) = j 1j(x 1 ) = a1x 1, the problem is reduced to finding a function X 1 ( x) converting the relationship (B.4) into the identity. Let us try to find the solution as an power series expansion (B.5) I 2 x =fj1x+f32x +... After substitution of this expression in equation (B.4) we obtain a1 (f3ix + fj2x 2 + ) = a1x + a2x 2 + 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 nonzero) 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 + b 2 x (a1 ~ -/= 0), the transformation y-l can be found as I 1 ( 2 2) x1 =XI + al b1x1 + b2x2 ; However a simple example (Maurin, 1989) shows that such a conclusion (equivalency of function to its first non-zero jet) is not valid

19 Appendix B: Certain Mathematical Notions of Catastrophe Theory 201 for the functions of more than one variable if its power expansion begins, 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- 1 j 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 can find an elementary transformation T 1-1 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 nonzero 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

20 202 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 = x 2, y = x 3, 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 = x 2 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.

21 Append ix B: Certain Mathematical Notions of Catastrophe Theory 203 Figure B.3. Replacing function y = xi - x +xi ~ + x ~ first non-zero jet (hyperboloid). at the point (0, 0) with its 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 = x 2, 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. Degenerate points are of the greatest interest from the viewpoint of qualitative change of behavior of different systems. However because small perturbations 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 simultaneous 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.

22 204 ASYMPTOTOLOGY: IDEAS, METHODS, AND APPLICATIONS Figure B.4. Replacing function y = x? + x +xi+ ~ x ~ first non-zero jet (paraboloid). at the point (0, 0) with its 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 coefficients 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. degenerate. 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 = x 3 and y = x 4. Certainly, the once-degenerate case will be the most frequently occurring because a change of control parameters, as a rule, leads to the disappearance 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

23 Appendix B: Certain Mathematical Notions of Catastrophe Theory 205 it is very difficult to arrange for threefold-degeneration. Also, for onceand 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 result 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 problem is the natural assumption that only "small perturbations" of the function at a degenerate critical point need to be considered. If a function 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 respect to non-degenerating coordinates as well as to products of degenerating and non-degenerating coordinates can be considered as negligibly small. At the same time the perturbations with respect to degenerating

24 206 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 perturbations 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 simplifications of the initial problem due to a determination of a small number of "dangerous" variables. Their behavior is described by a small number of nonlinear algebraic equations containing a few essential control parameters.

25 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 + Y 3 - kxy = 0 ( C.l) with an arbitrary value of k. After the transformations X=Jf3 1 X* Y==6f3 2 Y* k=jf3 3 k* ', (C.2) the terms of ( C.l) have powers of 6 as coefficients. Write down the corresponding 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 equation (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 Now introduce new variables X = P X*, Y == PY*, k = P k*. (C.6) X y X= k' y = k (C.7) 207