Localised nontopological structures: construction of solutions and stability problems

Possible methods are discussed for describing structures localised in finite region (solitons, vortices, defects and so on) within the framework of both integrable and nonintegrable field models. For integrable models a universal algorithm for the construction of soliton-like solutions is described and discussed in detail. This algorithm can be generalised to many-dimensional cases and its efficacy for several examples exceeds that of the standard inverse scattering transform method. For nonintegrable models we focus mainly on methods of studying the stability of soliton-like solutions, since stability problems become essential when one turns to a description of many-dimensional solitons. Special attention is paid to those stable localised structures that are not endowed with topological invariants, since for topologically nontrivial structures there exist effective methods of stability analysis, based on energy estimates. Here the principal topic is that of Lyapunov's direct method as applied to distributed systems are discussed. Effective stability criteria for stationary solitons, endowed with one or more charges, (the Q-theorem) are derived. Several examples are presented that illustrate the applicability of the method of functional estimates, and the stability of plasma solitons of the electron phase hole type is discussed.