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Numerical simulation of focusing stochastic nonlinear Schrödinger equations. (English) Zbl 0988.35156

Summary: 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.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35R60 PDEs with randomness, stochastic partial differential equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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