Numerical simulation of focusing stochastic nonlinear Schrِdinger equations

In this paper, we numerically investigate nonlinear Schrödinger equations with a stochastic contribution which is of white noise type and acts either as a potential (multiplicative noise) or as a forcing term (additive noise). In the subcritical case, we recover similar results as in the case of the Korteweg–de Vries equation. In the critical or supercritical case, we observe that depending on its smoothness, the noise may have different effects. Spatially smooth noises amplify blow-up phenomena, whereas delta correlated multiplicative noises prevent blow-up formation. Note that in this latter case, very few results are known, both from a theoretical and a numerical point of view.