Numerical resolution of stochastic focusing NLS equations

In this note, we numerically investigate a stochastic nonlinear Schrödinger equation derived as a perturbation of the deterministic NLS equation. The classical NLS equation with focusing nonlinearity of power law type is perturbed by a random term; it is a strong perturbation since we consider a space-time white noise. It acts either as a forcing term (additive noise) or as a potential (multiplicative noise). For simulations made on a uniform grid, we see that all trajectories blow-up in finite time, no matter how the initial data are chosen. Such a grid cannot represent a noise with zero correlation lengths, so that in these experiments, the noise is, in fact, spatially smooth. On the contrary, we simulate a noise with arbitrarily small scales using local refinement and show that in the multiplicative case, blow-up is prevented by a space-time white noise. We also present results on noise induced soliton diffusion.