Weakly nonlinear stochastic CGL equations

Kuksin, Sergei B.

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 4, pp. 1033-1056.

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:

$$\frac{\mathrm{d}}{\mathrm{d}t}u+\mathrm{i}\left(-\Delta +V\left(x\right)\right)u=\nu \left(\Delta u-{\gamma }_R{\left|u\right|}^{2p}u-\mathrm{i}{\gamma }_I{\left|u\right|}^{2q}u\right)+\sqrt{\nu }\eta (t,x).(*)$$

La force $\eta$ est un processus aléatoire blanc en temps $t$ et lisse en $x$; le potentiel $V\left(x\right)$ est typique. Nous étudions le comportement asymptotique des solutions sur de longs intervalles de temps $0\le t\le {\nu }^{-1}T$, quand $\nu \to 0$, et le comportement des solutions quand $t\to \infty$ et $\nu \to 0$. 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 $-\mathrm{i}{\gamma }_I{\left|u\right|}^{2q}u$ (mais elles dépendent de la partie dissipative $-{\gamma }_R{\left|u\right|}^{2p}u$). Si $p$ est un entier, on peut écrire ces équations explicitement.

We consider the linear Schrödinger equation under periodic boundary conditions, driven by a random force and damped by a quasilinear damping:

$$\frac{\mathrm{d}}{\mathrm{d}t}u+\mathrm{i}\left(-\Delta +V\left(x\right)\right)u=\nu \left(\Delta u-{\gamma }_R{\left|u\right|}^{2p}u-\mathrm{i}{\gamma }_I{\left|u\right|}^{2q}u\right)+\sqrt{\nu }\eta (t,x).(*)$$

The force $\eta$ is white in time and smooth in $x$; the potential $V\left(x\right)$ is typical. We are concerned with the limiting, as $\nu \to 0$, behaviour of solutions on long time-intervals $0\le t\le {\nu }^{-1}T$, and with behaviour of these solutions under the double limit $t\to \infty$ and $\nu \to 0$. 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 $-\mathrm{i}{\gamma }_I{\left|u\right|}^{2q}u$ (but depend on the dissipative part $-{\gamma }_R{\left|u\right|}^{2p}u$). If $p$ is an integer, they may be written explicitly.

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