LOCALIZED NONTOPOLOGICAL STRUCTURES - CONSTRUCTION OF SOLUTIONS AND STABILITY PROBLEMS

Possibilities in the description of structures localized in a finite region (solitons, vortices, defects and so on) within the framework of both integrable and nonintegrable field models are discussed. For integrable models a universal algorithm for the construction of soliton-like solutions is discussed in detail. This might be generalized to many dimensional cases and its efficacy for several examples exceeds that of the stadart inverse scattering transform method. For nonintegrable models we focus mainly on methods of studying the stability of soliton-like solutions, since stability problems become the most substantial ones, when one turns to a description of many dimensional solitons. We pay a special attention to those stable localized structures, which are not endowed with topological invariants, since for topologically nontrivial structures there exist efficacious method stability analysis, based on energy estimates. Here the principal topics of the Lyapunov's direct method in application to distributed systems are discussed. Efficacious criteria of stability for stationary the method of functional estimates and discuss the stability of plasma solitons of the electron phase hole type.