Dark spatial soliton break-up in the transverse plane

We report on numerical simulations investigating the instabilities which arise when dark soliton solutions of the two-dimensional (2D) nonlinear Schrodinger equation are allowed to propagate in a 3D self-defocussing medium. Firstly, propagation of fully 3D gaussians beams is studied. Then, on plane background, small sinusoidal and random perturbations are considered. We demonstrate break-up of the dark soliton profile into patterns of dark spots which are identified as phase singularities.