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Solitons, nonlinear evolution equations and inverse scattering. (English) Zbl 0762.35001
London Mathematical Society Lecture Note Series. 149. Cambridge (UK) etc.: Cambridge University Press. xii, 516 p. (1991).

The authors are well known mathematicians and herein they present a modern (from the beginning of the eighties) point of view on the applications of the inverse scattering transform method (ISTM) to different classes of equations in mathematical physics. A large bibliography is included and this book is an encyclopaedical source of information about the applications of the ISTM in the case of infinite interval of variables (the periodic boundary conditions are not considered). A lot of multidimensional equations are considered. The main tools for solving the corresponding inverse spectral problems are the Riemann-Hilbert and the “$D$-bar” $\left(\overline{\partial }\right)$ methods.

In chapter two the ISTM technique is applied to the KdV equations. In chapter three the inverse scattering associated with $N×N$ system is discussed. Chapter four details the ISTM to some integro-differential equations. Chapter five involves the $\overline{\partial }$-method and discusses a lot of $2+1$ dimensional equations. In chapter six the $\overline{\partial }$-method is applied to some $n+1$-dimensional equations and some recent results concerning the self-dual Yang-Mills equations are explained. In chapter seven the Painlevé equations, Painlevé property and Painlevé test are discussed. Finally some important open problems are stated.

This book is a “state of the art” explanation of some parts of the soliton theory from nowadays point of view.

MSC:
 35-01 Textbooks (partial differential equations) 35Q51 Soliton-like equations 37J35 Completely integrable systems, topological structure of phase space, integration methods 37K10 Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies 35Q53 KdV-like (Korteweg-de Vries) equations 58J72 Correspondences and other transformation methods (PDE on manifolds) 35Q15 Riemann-Hilbert problems 35Q58 Other completely integrable PDE (MSC2000) 35R30 Inverse problems for PDE 35Q55 NLS-like (nonlinear Schrödinger) equations 35P25 Scattering theory (PDE)