Solitons, nonlinear evolution equations and inverse scattering. (English) Zbl 0762.35001

Ablowitz, M. J.; Clarkson, P. A.

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” \((\overline\partial)\) methods.
In chapter two the ISTM technique is applied to the KdV equations. In chapter three the inverse scattering associated with \(N\times N\) 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.

Reviewer: Y.P.Mishev (Sofia)

MSC:

35-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
35Q51	Soliton equations
37J35	Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10	Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q53	KdV equations (Korteweg-de Vries equations)
58J72	Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds
35Q15	Riemann-Hilbert problems in context of PDEs
35Q58	Other completely integrable PDE (MSC2000)
35R30	Inverse problems for PDEs
35Q55	NLS equations (nonlinear Schrödinger equations)
35P25	Scattering theory for PDEs

Keywords:

inverse scattering transform method; delta-bar-method; inverse spectral problems; KdV equations; Yang-Mills equations; Painlevé equations; Painlevé test