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Zbl 0762.35001
Ablowitz, M.J.; Clarkson, P.A.
Solitons, nonlinear evolution equations and inverse scattering. (English)
[B] London Mathematical Society Lecture Note Series. 149. Cambridge (UK) etc.: Cambridge University Press. xii, 516 p. (1991). ISBN 0-521-38730-2/pbk

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.\par 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.\par This book is a ``state of the art'' explanation of some parts of the soliton theory from nowadays point of view.
[Y.P.Mishev (Sofia)]

MSC 2000:: *35-01 Textbooks (partial differential equations)
35Q51 Solitons
37J35 Completely integrable systems, etc.
37K10 Completely integrable systems etc.
35Q53 KdV-like equations
58J72 Correspondences and other transformation methods
35Q15 Riemann-Hilbert problems
35Q58 Other completely integrable PDE
35R30 Inverse problems for PDE
35Q55 NLS-like (nonlinear Schroedinger) equations
35P25 Scattering theory (PDE)

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

Cited in: Zbl 1232.35002 Zbl 1002.34001 Zbl 0994.34077 Zbl 0980.34085 Zbl 0962.34068 Zbl 0958.35131 Zbl 0939.37039 Zbl 0836.35002

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