# Time-Averaging for Weakly Nonlinear CGL Equations with Arbitrary Potentials

• Guan Huang
• Sergei Kuksin
• Alberto Maiocchi
Chapter
Part of the Fields Institute Communications book series (FIC, volume 75)

## Abstract

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.

## Notes

### Acknowledgements

We are thankful to Anatoli Neishtadt for discussing the finite-dimensional averaging. This work was supported by l’Agence Nationale de la Recherche through the grant STOSYMAP (ANR 2011BS0101501).

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