Time-averaging forweakly nonlinear CGL equations with arbitrary potentials

Consider weakly nonlinear complex Ginzburg–Landau (CGL) equation of the form: u_t + i(-Δu + V(x)u) = ε μΔu + ε(∇u, u), x ∈ R^d ; (*) under the periodic boundary conditions, where μ ≥0 and P is a smooth function. Let {ζ₁ (x), ζ₂(x); : : : } be the L₂-basis formed by eigenfunctions of the operator -Δ + V(x). For a complex function u(x), write it as u(x)= ∑_{k ≥1} v_kζ_k(x) and set I_k (u) = 1/2 |v_k|². Then for any solution u(t, x) of the linear equation (*)_ε=0we 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 ε^-1, then for any its solution u^ε(t, x), the limiting behavior of the curve I.u^ε(t,.)) on time intervals of order "ε^-1, as " ε → 0, can be uniquely characterized by a solution of a certain well-posed effective equation: u_t =ε μ Δ Dμ + ε F(u); where F(u), is a resonant averaging of the nonlinearity P(∇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 √ε 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.