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Solitary waves in media with dispersion and dissipation. (A review). (English. Russian original) Zbl 0995.76013
Fluid Dyn. 35, No. 2, 157-176 (2000); translation from Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza 2000, No. 2, 3-27 (2000).
Summary: We review the latest results on the theory of nonlinear waves in dispersive and dissipative media. Attention is concentrated on small-amplitude solitary waves and, in particular, on the classification of types of solitary waves, conditions for their existence, the evolution of local perturbations associated with the presence of solitarv waves of various types, and 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.

MSC:
76B25 Solitary waves for incompressible inviscid fluids
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
35Q51 Soliton equations
35Q53 KdV equations (Korteweg-de Vries equations)
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