Numerical study of two-dimensional stochastic NLS equations

In this article, we numerically solve the two-dimensional stochastic nonlinear Schrodinger equation in the case of multiplicative and additive white noises. The aim is to investigate their influence on well-known deterministic solutions: stationary states and blowing-up solutions. In the first case, we find that a multiplicative noise has a damping effect very similar to diffusion. However, for small amplitudes of the noise, the structure of solitary state is still localized. In the second case, a local refinement algorithm is used to overcome the difficulty arising for the computation of singular solutions. Our experiments show that multiplicative white noise stops the deterministic blow-up that occurs in the critical case. This extends the results of Debussche and Di Menza (Physica D, 162(3-4) 2002, 131-154) in the one-dimensional case. (c) 2005 Wiley Periodicals, Inc.