Weakly nonlinear stochastic CGL equations
Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 4, p. 1033-1056

We consider the linear Schrödinger equation under periodic boundary conditions, driven by a random force and damped by a quasilinear damping: d dtu+i- Δ + V ( x )u=νΔ u - γ R |u| 2p u - i γ I |u| 2q u+νη(t,x).(*) The force η is white in time and smooth in x; the potential V(x) is typical. We are concerned with the limiting, as ν0, behaviour of solutions on long time-intervals 0tν -1 T, and with behaviour of these solutions under the double limit t and ν0. We show that these two limiting behaviours may be described in terms of solutions for the system 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 -iγ I |u| 2q u (but depend on the dissipative part -γ R |u| 2p u). If p is an integer, they may be written explicitly.

Nous considérons l’équation de Schrödinger linéaire avec les conditions aux limites périodiques, perturbée par une force aléatoire et amortie par un terme quasi linéaire: d dtu+i- Δ + V ( x )u=νΔ u - γ R |u| 2p u - i γ I |u| 2q u+νη(t,x).(*) La force η est un processus aléatoire blanc en temps t et lisse en x; le potentiel V(x) est typique. Nous étudions le comportement asymptotique des solutions sur de longs intervalles de temps 0tν -1 T, quand ν0, et le comportement des solutions quand t et ν0. Nous démontrons qu’on peut décrire ces deux comportements asymptotiques en termes des solutions du système d'équations effectives pour (*). Ce dernier est une équation de la chaleur avec un terme quasi linéaire non local et une force aléatoire lisse additive, qui est écrite dans l’espace de Fourier. Les équations ne dépendent pas de la partie hamiltonienne de la perturbation -iγ I |u| 2q u (mais elles dépendent de la partie dissipative -γ R |u| 2p u). Si p est un entier, on peut écrire ces équations explicitement.

DOI : https://doi.org/10.1214/11-AIHP482
Classification:  35Q56,  60H15
Keywords: complex Ginzburg-Landau equation, small nonlinearity, stationary measures, averaging, effective equations
@article{AIHPB_2013__49_4_1033_0,
     author = {Kuksin, Sergei B.},
     title = {Weakly nonlinear stochastic CGL equations},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {49},
     number = {4},
     year = {2013},
     pages = {1033-1056},
     doi = {10.1214/11-AIHP482},
     zbl = {1280.35144},
     mrnumber = {3127912},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2013__49_4_1033_0}
}
Weakly nonlinear stochastic CGL equations. Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 4, pp. 1033-1056. doi : 10.1214/11-AIHP482. http://www.numdam.org/item/AIHPB_2013__49_4_1033_0/

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