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  • Journées équations aux dérivées partielles
  • Year 2001
  • article no. 3
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Theoretical and numerical aspects of stochastic nonlinear Schrödinger equations
Anne de Bouard; Arnaud Debussche; Laurent Di Menza
Journées équations aux dérivées partielles (2001), article no. 3, 13 p.
  • Abstract

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.

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MR   Zbl
DOI: 10.5802/jedp.587
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@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
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%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/
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