Weakly nonlinear stochastic CGL equations

Abstract

We consider the linear Schrödinger equation under periodic boundary condition, driven by a random force and damped by a quasilinear damping: $$ \frac{d}{dt}u+i\big(-Δ+V(x)\big) u=ν\Big(Δu-\gr |u|^{2p}u-i\gi |u|^{2q}u \Big) +\sqrtν\, η(t,x).\qquad (*) $$ The force $\eta$ is white in time and smooth in $x$. We are concerned with the limiting, as $\nu \to 0$, behaviour of its 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 {\it system of effective equations for $(\ast )$} 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 $-i\gi|u|^{2q}u$ (but depend on the dissipative part $-\gr|u|^{2p}u$). If $p$ is an integer, they may be written explicitly.