Self-focusing and transverse instabilities of solitary waves

We give an overview of the basic physical concepts and analytical methods for 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 modulational (induced by temporal wave dispersion) instabilities, in the framework of the cubic nonlinear Schrödinger (NLS) equation and its generalizations. Both linear and nonlinear regimes of the instability-induced soliton dynamics are analyzed for bright (self-focusing media) and dark (self-defocusing media) solitary waves. For a defocusing Kerr medium, the results of the small-amplitude limit are compared with the theory of the transverse instabilities of the Korteweg–de Vries solitons developed in the framework of the exactly integrable Kadomtsev–Petviashvili equation. We give also a comprehensive summary of different physical problems involving the analysis of the transverse and modulational instabilities of solitary waves including the soliton self-focusing in the discrete NLS equation, the models of parametric wave mixing, the Davey–Stewartson equation, the Zakharov–Kuznetsov and Shrira equations, instabilities of higher-order and ring-like spatially localized modes, the kink stability in the dissipative Cahn–Hilliard equation, etc. Experimental observations of the soliton self-focusing and transverse instabilities for bright and dark solitons in nonlinear optics are briefly summarized as well.