Introduction

Abstract

During the last 30 years the theory of solitons - the theory of nonlinear partial differential equations (PDEs) possessing solutions of a special kind — has grown into a large field that attracts the attention of both mathematicians and physicists in view of its important applications and of the novelty of the problems. Physical problems leading to the equations under consideration are observed, for example, in the monograph by V.G. Makhankov [60]. One of the related mathematical discoveries is the possibility of studying certain nonlinear equations from this field by methods that were developed to analyze the quantum inverse scattering problem; these equations are called solvable by the method of the inverse scattering problem (on this subject, see, for example [89,94]). At the same time, the class of currently known nonlinear PDEs solvable by this method is sufficiently narrow and, on the other hand, there is another approach, called the qualitative theory of differential equations. The latter approach in particular includes investigations on the well-posedness of various problems for these equations, the behavior of solutions such as stability or blowing-up, properties of dynamical systems generated by these equations, etc., and this approach makes it possible to investigate an essentially wider class of problems (maybe in a more general study).