Self-focusing of plane dark solitons in nonlinear defocusing media

We analyze a transverse instability of plane (quasi-one-dimensional) dark solitons in the framework of the two-dimensional nonlinear Schrödinger (NLS) equation for beam propagation in a defocusing nonlinear medium. We show that in the vicinity of the instability threshold the exponential growth of transverse perturbations is stabilized by nonlinearity and also by the radiation emitted from the plane dark soliton to the right and left. Dynamics of the transverse instability of the plane dark soliton of arbitrary amplitude is investigated analytically by means of the asymptotic technique, and also numerically by direct integration of the two-dimensional NLS equation. In particular we show that there exist generally three different scenarios of the instability dynamics, namely, (i) generation of a chain of two-dimensional ‘‘gray’’ solitons (anisotropic solitons of the Kadomtsev-Petviashvili, or KP1, equation) from the small-amplitude plane dark soliton, (ii) long-lived large-amplitude transverse oscillations of the plane dark soliton near the instability threshold, and finally, (iii) decay of the plane dark soliton into a chain of circular symmetric ‘‘black’’ solitons (optical vortices) of alternative topological charges. We estimate the region of the instability domain for the parameters of the soliton and perturbation where the instability of the plane dark soliton ends up in the formation of pairs of vortex and antivortex solitons.