Theoretical and numerical aspects of stochastic nonlinear Schrödinger equations

de Bouard, Anne, Debussche, Arnaud, and Di Menza, Laurent. "Theoretical and numerical aspects of stochastic nonlinear Schrödinger equations." Journées équations aux dérivées partielles (2001): 1-13. <http://eudml.org/doc/93414>.

@article{deBouard2001,
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.},
author = {de Bouard, Anne, Debussche, Arnaud, Di Menza, Laurent},
journal = {Journées équations aux dérivées partielles},
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},
language = {eng},
pages = {1-13},
publisher = {Université de Nantes},
title = {Theoretical and numerical aspects of stochastic nonlinear Schrödinger equations},
url = {http://eudml.org/doc/93414},
year = {2001},
}

TY - JOUR
AU - de Bouard, Anne
AU - Debussche, Arnaud
AU - Di Menza, Laurent
TI - Theoretical and numerical aspects of stochastic nonlinear Schrödinger equations
JO - Journées équations aux dérivées partielles
PY - 2001
PB - Université de Nantes
SP - 1
EP - 13
AB - 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.
LA - eng
KW - 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
UR - http://eudml.org/doc/93414
ER -