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In the last two decades a large number of investigations in different areas of physics have been devoted to the study of nonlinear wave processes, for instance, various questions relating to the theory of plasma and nonlinear optics (except the classical problems of hydrodynamics). Simple'model'nonlinear wave equations were constructed in the course of the development of the nonlinear wave theory; in some sense, these equations are universal, ie, they may be encountered, just like the classical d'Alembert linear equation, in diverse physical problems. Examples of such equations are the Kortweg-de Vries equation (KdV), the nonlinear Schrödinger equation, and the sine-Gordon equation. These equations exhibit, at least in the one-dimensional case, a remarkable mathematical property. They possess hidden algebraic symmetry, as a result of which they can be'integrated'by the so-called inverse method for an auxiliary linear operator. This book is largely concerned with this method and its generalization, the main purpose being to give an elementary, as far as possible, presentation of this method and all the necessary preliminaries from the theory of scattering, Riemann surfaces, Hamiltonian systems and others. A special chapter is devoted to the asymptotic behavior of solutions over large time intervals. In certain cases important qualitative results may be obtained by means of the methods that are not related to the scattering theory (see § 4, Chapter IV) but resemble the classical'averaging method'of Bogolyubov and others.