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Theory of solitons. The method of the inverse problem. (Теория солитонов. Метод обратной задачи.) (Russian) Zbl 0598.35003

Moskva: Izdatel’stvo “Nauka”. 320 p. R. 1.90 (1980).
This is the first book on the famous inverse scattering method. The authors are well-known physicists and mathematicians who have made a number of crucial contributions to the subject. Let me sketch the idea of the inverse scattering method on the example of the KdV equation \[ u_ t=uu_ x+u_{xxx}.\tag{1} \] One associates with the solution \(u(x,t)\) of this nonlinear evolution equation the Schrödinger operator \(L(t)=-D^ 2+u\) with the time-dependent potential \(u(\cdot,t)\). It turns out that while \(u(\cdot,t)\) evolves in time according to (1), the scattering coefficients \(a(k,t), b(k,t)\) of \(L(t)\) evolve according to linear equations \[ a_ t=0, \;b_ t=8\sqrt{-1} k^ 3b.\tag{2} \] Since the correspondence \(L\to (a,b)\) is one-to-one (not quite but let us pretend it is so), we may view the inverse scattering method as a change of variables \(u\to (a,b)\) which transforms the nonlinear KdV equation (1) into a system of linear equations (2).
In the introduction the authors show that weakly nonlinear evolution processes are described by equations of type (1). This explains why KdV and its cousins (sine-Gordon, nonlinear Schrödinger, Kadomtsev-Petviashvili, etc.) arise in various physical contexts. The first chapter of the book is a detailed exposition of the inverse scattering method on the example of KdV. Other examples are treated as well. All relevant facts from the scattering theory are explained in the text. The authors also discuss some related subjects like integrals of motion, KdV as an integrable Hamiltonian system, integrable partial difference equations (Toda lattice), etc.
The “classical” inverse scattering method describes the space-decaying solutions of equations of KdV type. Its generalization to the periodic and quasiperiodic solutions required completely new ideas and was created as recently as in 1975–1976. This material is the subject of Chapter 2.
In Chapter 3 the authors apply the previous integration techniques to the Riemann problem and to some relativistic field equations. Chapter 4 is about the time asymptotics of solutions of the equations of KdV type.
The Appendix briefly treats a generalization of the inverse scattering method to the case of more than one space variable. The working example here is the Kadomtsev-Petviashvili equation (two-dimensional KdV).
The book requires very little preliminary knowledge from the reader. It includes pertinent material from such far-apart subjects as scattering theory and Riemann surfaces. The exposition is very informal, which is a big advantage for the reader learning such an immense subject as the inverse scattering method.
An English translation has appeared, see Zbl 0598.35002.

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
35Q51 Soliton equations
35Q53 KdV equations (Korteweg-de Vries equations)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
35P25 Scattering theory for PDEs