Weakly nonlinear stochastic CGL equations

2011/06/06 by Kuksin, Sergei B.
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1106.1158

Abstract

We consider the linear Schrödinger equation under periodic boundary condition, driven by a random force and damped by a quasilinear damping: (d)/(dt)u+i(-Δ+V(x)) u=ν(Δu-\gr |u|^2pu-i\gi |u|^2qu ) +√ν η(t,x). (*) The force η is white in time and smooth in x. We are concerned with the limiting, as ν→0, behaviour of its solutions on long time-intervals 0≤ t≤ν^-1T, and with behaviour of these solutions under the double limit t→∞ and ν→0. We show that these two limiting behaviours may be described in terms of solutions for the \it 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 -i\gi|u|^2qu (but depend on the dissipative part -\gr|u|^2pu). If p is an integer, they may be written explicitly.

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