We describe several results obtained recently on stochastic nonlinear Schrödinger equations. We show that under suitable smoothness assumptions on the noise, the nonlinear Schrödinger 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.
@article{JEDP_2001____A3_0,
author = {de Bouard, Anne and Debussche, Arnaud and Di Menza, Laurent},
title = {Theoretical and numerical aspects of stochastic nonlinear Schr\"odinger equations},
journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
year = {2001},
pages = {1-13},
doi = {10.5802/jedp.587},
mrnumber = {1843404},
zbl = {1005.35084},
language = {en},
url = {http://dml.mathdoc.fr/item/JEDP_2001____A3_0}
}
de Bouard, Anne; Debussche, Arnaud; Di Menza, Laurent. Theoretical and numerical aspects of stochastic nonlinear Schrödinger equations. Journées équations aux dérivées partielles, (2001), pp. 1-13. doi : 10.5802/jedp.587. http://gdmltest.u-ga.fr/item/JEDP_2001____A3_0/