Time-averaging forweakly nonlinear CGL equations with arbitrary potentials

Consider weakly nonlinear complex Ginzburg-Landau (CGL) equation of the form: under the periodic boundary conditions, where u ≥ 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) = Σk≥1 ν_kζk(x) and set I_k(u) = 1/2 |v_k|². Then for any solution u(t,x) of the linear equation .(*)ε=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 ≲ ε^-1, then for any its solution uε(t, x), the limiting behavior of the curve I(uε(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: where F(u) is a resonant averaging of the nonlinearity P(Δu, u). We also prove similar results for the stochastically perturbed equation, when a white in time and in x random force of orderp 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.