Numerical study of two dimensional stochastic NLS equations
Résumé
In this paper, we numerically solve the two-dimensional stochastic nonlinear Shrödinger 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 expirements show that multiplicative white noise stops the deterministic blow-up which occurs in the critical case. This extends the results of [15] in the one-dimensional case.