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.
@incollection{JEDP_2001____A3_0, author = {Anne de Bouard and Arnaud Debussche and Laurent Di Menza}, title = {Theoretical and numerical aspects of stochastic nonlinear {Schr\"odinger} equations}, booktitle = {}, series = {Journ\'ees \'equations aux d\'eriv\'ees partielles}, eid = {3}, pages = {1--13}, publisher = {Universit\'e de Nantes}, year = {2001}, doi = {10.5802/jedp.587}, zbl = {1005.35084}, mrnumber = {1843404}, language = {en}, url = {https://proceedings.centre-mersenne.org/articles/10.5802/jedp.587/} }
TY - JOUR AU - Anne de Bouard AU - Arnaud Debussche AU - Laurent Di Menza TI - Theoretical and numerical aspects of stochastic nonlinear Schrödinger equations JO - Journées équations aux dérivées partielles PY - 2001 SP - 1 EP - 13 PB - Université de Nantes UR - https://proceedings.centre-mersenne.org/articles/10.5802/jedp.587/ DO - 10.5802/jedp.587 LA - en ID - JEDP_2001____A3_0 ER -
%0 Journal Article %A Anne de Bouard %A Arnaud Debussche %A Laurent Di Menza %T Theoretical and numerical aspects of stochastic nonlinear Schrödinger equations %J Journées équations aux dérivées partielles %D 2001 %P 1-13 %I Université de Nantes %U https://proceedings.centre-mersenne.org/articles/10.5802/jedp.587/ %R 10.5802/jedp.587 %G en %F JEDP_2001____A3_0
Anne de Bouard; Arnaud Debussche; Laurent Di Menza. Theoretical and numerical aspects of stochastic nonlinear Schrödinger equations. Journées équations aux dérivées partielles (2001), article no. 3, 13 p. doi : 10.5802/jedp.587. https://proceedings.centre-mersenne.org/articles/10.5802/jedp.587/
Cited by Sources: