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Consider weakly nonlinear complex Ginzburg–Landau (CGL) equation of the form:  $$\displaystyle{ u_{t} + i(-\bigtriangleup u + V (x)u) =\varepsilon \mu \varDelta u +\varepsilon \mathcal{P}(\nabla u,u),\quad x \in \mathbb{R}^{d}\,, }$$  (*)  under the periodic boundary conditions, where μ ≥ 0 and $\mathcal{P}$ is a smooth function. Let $\{\zeta _{1}(x),\zeta _{2}(x),\ldots \}$ be the L2-basis formed by eigenfunctions of the operator −△ + V (x). For a complex function u(x), write it as u(x) = ∑k ≥ 1v k ζ k (x) and set $I_{k}(u) = \frac{1} {2}\vert v_{k}\vert ^{2}$. Then for any solution u(t, x) of the linear equation $({\ast})_{\varepsilon =0}$ we have I(u(t, ⋅ )) = const. In this work it is proved that if equation (∗) with a sufficiently smooth real potential V (x) is well posed on time-intervals $t \lesssim \varepsilon ^{-1}$, then for any its solution $u^{\varepsilon }(t,x)$, the limiting behavior of the curve $I(u^{\varepsilon }(t,\cdot ))$ on time intervals of order $\varepsilon ^{-1}$, as $\varepsilon \rightarrow 0$, can be uniquely characterized by a solution of a certain well-posed effective equation:  $$\displaystyle{u_{t} =\varepsilon \mu \bigtriangleup u +\varepsilon F(u),}$$  where F(u) is a resonant averaging of the nonlinearity $\mathcal{P}(\nabla u,u)$. We also prove similar results for the stochastically perturbed equation, when a white in time and smooth in x random force of order $\sqrt{\varepsilon }$ 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 $\mathbb{R}^{d}$ under Dirichlet boundary conditions.

Time-Averaging for Weakly Nonlinear CGL Equations with Arbitrary Potentials