Solitary waves in media with dispersion and dissipation (a review)

Abstract

The latest results relating to the theory of nonlinear waves in dispersive and dissipative media are reviewed. Attention is concentrated on small-amplitude solitary waves and, in particular, on the classification of types of solitary waves, their conditions of existence, the evolution of local perturbations associated with the presence of solitary waves of various types, and problems of the existence of nonlinear waves localized with respect to a particular direction as the space dimension increases (spontaneous dimension breaking). As examples of dispersive and dissipative media admitting plane solitary waves of various types, we consider a cold collisionless plasma, an ideal incompressible fluid of finite depth beneath an elastic plate and with surface tension, and a fluid in a rapidly oscillating rectangular vessel (Faraday resonance). Examples of spontaneous dimension breaking are considered for the generalized Kadomtsev-Petviashvili equation.