Time-averaging for weakly nonlinear CGL equations with arbitrary potentials

Abstract

Consider weakly nonlinear complex Ginzburg--Landau (CGL) equation of the form: $$ u_t+i(-\Delta u+V(x)u)=\epsilon\mu\Delta u+\epsilon \mathcal{P}( u),\quad x\in {R^d}\,, \quad(*) $$ under the periodic boundary conditions, where $\mu\geqslant0$ and $\mathcal{P}$ is a smooth function. Let $\{\zeta_1(x),\zeta_2(x),\dots\}$ be the $L_2$-basis formed by eigenfunctions of the operator $-\Delta +V(x)$. For a complex function $u(x)$, write it as $u(x)=\sum_{k\geqslant1}v_k\zeta_k(x)$ and set $I_k(u)=\frac{1}{2}|v_k|^2$. Then for any solution $u(t,x)$ of the linear equation $(*)_{\epsilon=0}$ we have $I(u(t,\cdot))=const$. In this work it is proved that if equation $(*)$ with a sufficiently smooth real potential $V(x)$ is well posed on time-intervals $t\lesssim \epsilon^{-1}$, then for any its solution $u^{\epsilon}(t,x)$, the limiting behavior of the curve $I(u^{\epsilon}(t,\cdot))$ on time intervals of order $\epsilon^{-1}$, as $\epsilon\to0$, can be uniquely characterized by a solution of a certain well-posed effective equation: $$ u_t=\epsilon\mu\triangle u+\epsilon F(u), $$ where $F(u)$ is a resonant averaging of the nonlinearity $\mathcal{P}(u)$. We also prove a similar results for the stochastically perturbed equation, when a white in time and smooth in $x$ random force of order $\sqrt\epsilon$ is added to the right-hand side of the equation. The approach of this work is rather general. In particular, it applies to equations in bounded domains in $R^d$ under Dirichlet boundary conditions.