Self-focusing and transverse instabilities of solitary waves

We give an overview of the basic physical concepts and analytical methods f or investigating the symmetry-breaking instabilities of solitary waves. We discuss self-focusing of spatial optical solitons in diffractive nonlinear media due to either transverse (one more unbounded spatial dimension) or mo dulational (induced by temporal wave dispersion) instabilities, in the fram ework of the cubic nonlinear Schrodinger (NLS) equation and its generalizat ions. Both linear and nonlinear regimes of the instability-induced soliton dynamics are analyzed for bright (self-focusing media) and dark (self-defoc using media) solitary waves. For a defocusing Kerr medium, the results of t he small-amplitude limit are compared with the theory of the transverse ins tabilities of the Korteweg-de Vries solitons developed in the framework of the exactly integrable Kadomtsev-Petviashvili equation. We give also a comp rehensive summary of different physical problems involving the analysis of the transverse and modulational instabilities of solitary waves including t he soliton self-focusing in the discrete NLS equation, the models of parame tric wave mixing, the Davey-Stewartson equation, the Zakharov-Kuznetsov and Shrira equations, instabilities of higher-order and ring-like spatially lo calized modes, the kink stability in the dissipative Cahn-Hilliard equation , etc. Experimental observations of the soliton self-focusing and transvers e instabilities for bright and dark solitons in nonlinear optics are briefl y summarized as well. (C) 2000 Elsevier Science B.V. All rights reserved.