Theoretical and numerical aspects of stochastic nonlinear Schrödinger equations. (English) Zbl 1005.35084

de Bouard, Anne; Debussche, Arnaud; Di Menza, Laurent

Journées “Équations aux dérivées partielles”, Plestin-les-Grèves, France, 5 au 8 juin 2001. Exposés Nos. I-XIV. Nantes: Université de Nantes. Exp. No. 3, 13 p. (2001).

Summary: We describe several results obtained recently on stochastic nonlinear Schrödinger equations. We show that under suitable smoothness assumptions on the noise, the nonlinear Schrödinger equation perturbed by an additive or multiplicative noise is well posed under similar assumptions on the nonlinear term as in the deterministic theory.
Then, we restrict our attention to the case of a focusing nonlinearity with critical or supercritical exponent. If the noise is additive, smooth in space and non degenerate, we prove that any initial data gives birth to a singular solution; thus the noise changes the qualitative behavior since, as is well known, in the deterministic case only a restricted class of initial data give a solution which blows up.
We also present numerical experiments which indicate that, on the contrary, a multiplicative white noise seems to prevent blow up. We finally give a convergence result for the numerical scheme used in these simulations.
For the entire collection see [Zbl 0990.00046].

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
35R60	PDEs with randomness, stochastic partial differential equations
60H15	Stochastic partial differential equations (aspects of stochastic analysis)
76B15	Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76M35	Stochastic analysis applied to problems in fluid mechanics

Keywords:

blow-up; white noise; finite difference scheme; mesh refinement; nonlinear Schrödinger equation perturbed by an additive or multiplicative noise; well posed; focusing nonlinearity; singular solution