Weakly nonlinear stochastic CGL equations

We consider the linear Schr"odinger equation under periodic boundary condition, driven by a random force and damped by a quasilinear damping: $$\frac{d}{dt}u+i(-\mathrm{\Delta }+V(x))u=\nu (\mathrm{\Delta }u-\text{gr}\mathrm{\mid }u{\mathrm{\mid }}^{2p}u-i\text{gi}\mathrm{\mid }u{\mathrm{\mid }}^{2q}u)+\sqrt{\nu }\eta (t,x)\mathrm{.}(\ast )$$ 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 $t\to \mathrm{\infty }$ 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 $-i\text{gi}\mathrm{\mid }u{\mathrm{\mid }}^{2q}u$ (but depend on the dissipative part $-\text{gr}\mathrm{\mid }u{\mathrm{\mid }}^{2p}u$). If $p$ is an integer, they may be written explicitly.