Collapse of solitary excitations in the nonlinear Schrödinger equation with nonlinear damping and white noise

Abstract

We study the effect of adding noise and nonlinear damping in the two-dimensional nonlinear Schr\"odinger equation (NLS). Using a collective coordinate approach, we find that for initial conditions where total collapse occurs in the unperturbed NLS, the presence of the damping term will instead result in an exponentially decreasing width of the solution in the long-time limit. We also find that a sufficiently large noise variance may cause an initially localized distribution to spread instead of contracting, and that the critical variance necessary to cause dispersion will for small damping be the same as for the undamped system. \textcopyright{} 1996 The American Physical Society.