There is a large class of nonlinear partial differential equations, including the famous KdV equation, which can be integrated by inverse scattering techniques and admit infinitely many local conservation laws, parametric Bäcklund transformations as well as solutions in terms of theta functions of algebraic curves. The present book aims at systematically discussing parts of the theory of these equations, the so-called soliton theory.
At present, there exists quite a variety of books treating this topic, among them P. G. Drazin [Solitons, London Mathematical Society, Lecture Notes Series 85 (1983; Zbl 0517.35002)] and the book of L. A. Takhtadzhyan and L. D. Faddeev [Hamiltonian methods in the theory of solitons, Nauka, Moskva (1986; Zbl 0632.58003)]. Going deeper into the analytic theory of solitons, compared to the first mentioned book, the present monograph of I. Cherednik also enhances the material covered in the second one, basically by the study of the algebraic aspects of the theory. However, the main feature with respect to the existing literature consists in a thorough investigation of general matrix soliton equations.
The text consists of an introduction and two chapters. The introduction provides an overview of the historical developement of the theory and the existing related literature. Furthermore, the author explains the basic problems and the key constructions, and he gives a list of related topics which are not treated in his book.
Chapter I is dedicated to conservation laws and algebro-geometric solutions of soliton equations, including generalized Lax equations, the theory of Baker functions and, as applications, algebro-geometric solutions of the sine-Gordon equation and the nonlinear Schrödinger equation.
The second chapter of the book provides a systematic account on Bäcklund transformations, as well as an introduction to scattering theory and an application of the inverse problem methods. High attention is paid to several concrete equations, e.g. the equation of chiral fields, the Heisenberg magnet, the sine-Gordon equation, the nonlinear Schrödinger equation, and others.
A continuation of this book [J. Cherednik, “Loop groups in soliton theory”, Chapters III–IV of “Algebraic methods of soliton theory” (Russian) (1994)] provides some material on loop groups in soliton theory. Assuming only a basic mathematical knowledge, the book guides the reader to the forefront of research in soliton theory. The respective sections are fairly independent such allowing the reader to study each particular subject without reading the entire book.
The arrangement of each section matches all needs of the reader. Each section comes with its own introduction which not only gives an outline of the contents of the section, but also precisely summarizes all prerequisites for reading, and that with detailed references. In the present text, all statements are proven completely. At the end of each section, detailed comments are added in order to explain the historical background and to give hints for further reading.
The author, one of the most active researchers in the field of solitons, presents in a skillful manner the modern methods of soliton theory and leads the reader to contemporary research developments. Therefore, the book is particularly useful for both specialists and researchers starting to work in the domain of solitons. However, in view of its comprehensive and methodically masterly style, the book may serve as an excellent textbook, as well.