Weakly nonlinear stochastic CGL equations

@article{Kuksin2013,
abstract = {We consider the linear Schrödinger equation under periodic boundary conditions, driven by a random force and damped by a quasilinear damping: \[\frac\{\mathrm \{d\}\}\{\mathrm \{d\}t\}u+\mathrm \{i\}\bigl (-\Delta +V(x)\bigr )u=\nu \bigl (\Delta u-\gamma \_\{R\}|u|^\{2p\}u-\mathrm \{i\}\gamma \_\{I\}|u|^\{2q\}u\bigr )+\sqrt\{\nu \}\eta (t,x).\quad (\ast )\] The force $\eta $ is white in time and smooth in $x$; the potential $V(x)$ is typical. We are concerned with the limiting, as $\nu \rightarrow 0$, behaviour of solutions on long time-intervals $0\le t\le \nu ^\{-1\}T$, and with behaviour of these solutions under the double limit $t\rightarrow \infty $ and $\nu \rightarrow 0$. We show that these two limiting behaviours may be described in terms of solutions for thesystem of effective equations for($*$) which is a well posed semilinear stochastic heat equation with a non-local nonlinearity and a smooth additive noise, written in Fourier coefficients. The effective equations do not depend on the Hamiltonian part of the perturbation $-\mathrm \{i\}\gamma _\{I\}|u|^\{2q\}u$ (but depend on the dissipative part $-\gamma _\{R\}|u|^\{2p\}u$). If $p$ is an integer, they may be written explicitly.},
author = {Kuksin, Sergei B.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {complex Ginzburg–Landau equation; small nonlinearity; stationary measures; averaging; effective equations; complex Ginzburg-Landau equation; limiting behaviour; stationary measure; effective equation},
language = {eng},
number = {4},
pages = {1033-1056},
publisher = {Gauthier-Villars},
title = {Weakly nonlinear stochastic CGL equations},
url = {http://eudml.org/doc/272023},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Kuksin, Sergei B.
TI - Weakly nonlinear stochastic CGL equations
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2013
PB - Gauthier-Villars
VL - 49
IS - 4
SP - 1033
EP - 1056
AB - We consider the linear Schrödinger equation under periodic boundary conditions, driven by a random force and damped by a quasilinear damping: \[\frac{\mathrm {d}}{\mathrm {d}t}u+\mathrm {i}\bigl (-\Delta +V(x)\bigr )u=\nu \bigl (\Delta u-\gamma _{R}|u|^{2p}u-\mathrm {i}\gamma _{I}|u|^{2q}u\bigr )+\sqrt{\nu }\eta (t,x).\quad (\ast )\] The force $\eta $ is white in time and smooth in $x$; the potential $V(x)$ is typical. We are concerned with the limiting, as $\nu \rightarrow 0$, behaviour of solutions on long time-intervals $0\le t\le \nu ^{-1}T$, and with behaviour of these solutions under the double limit $t\rightarrow \infty $ and $\nu \rightarrow 0$. We show that these two limiting behaviours may be described in terms of solutions for thesystem of effective equations for($*$) which is a well posed semilinear stochastic heat equation with a non-local nonlinearity and a smooth additive noise, written in Fourier coefficients. The effective equations do not depend on the Hamiltonian part of the perturbation $-\mathrm {i}\gamma _{I}|u|^{2q}u$ (but depend on the dissipative part $-\gamma _{R}|u|^{2p}u$). If $p$ is an integer, they may be written explicitly.
LA - eng
KW - complex Ginzburg–Landau equation; small nonlinearity; stationary measures; averaging; effective equations; complex Ginzburg-Landau equation; limiting behaviour; stationary measure; effective equation
UR - http://eudml.org/doc/272023
ER -