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

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

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