Time-Averaging for Weakly Nonlinear CGL Equations with Arbitrary Potentials

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 L ₂-basis formed by eigenfunctions of the operator −△ + V (x). For a complex function u(x), write it as u(x) = ∑ _k ≥ 1 v _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.

Dedicated to Walter Craig on his 60th birthday

Appendix

Consider the CGL equation (14), where $\mathcal{P}: \mathbb{C}^{d(d+1)/2+d+1} \times T^{d} \rightarrow \mathbb{C}$ is a $C^{\infty }$-smooth function. We write it in the v-variables and slow time $\tau =\varepsilon t$:

$$\displaystyle{\dot{v}_{k} +\varepsilon ^{-1}i\lambda _{ k}v_{k} = P_{k}(v),\quad k \in \mathbb{N},}$$

where

$$\displaystyle{P(v):= (P_{k}(v),k \in \mathbb{N}) =\varPsi (\mathcal{P}(\nabla ^{2}u,\nabla u,u,x)),\;\;\;u =\varPsi ^{-1}v,}$$

and introduce the effective equation

$$\displaystyle{ \dot{\tilde{a}} =\langle P\rangle _{\varLambda }(\tilde{a}). }$$

(56)

By Lemma 3 P defines smooth locally Lipschitz mappings h ^s → h ^s−2 for s > 2 + d∕2. So by a version of Lemma 4, $\langle P\rangle _{\varLambda } \in Lip_{loc}(h^{s};h^{s-2})$ for s > 2 + d∕2. Assume that

Assumption E.

There exists s ₀ ∈ (d∕2,n] such that the effective equation (56) is locally well posed in the Hilbert spaces h ^s , with $s \in [s_{0},n] \cap \mathbb{N}$ .

Let $u^{\varepsilon }(t,x)$ be a solution of Eq. (14) with initial datum u ₀ ∈ H ^s, $v^{\varepsilon }(\tau ) =\varPsi (u(\varepsilon ^{-1}\tau,x))$, and $\tilde{a}(\tau )$ be a solution of Eq. (56) with initial datum Ψ(u ₀). Then we have the following result:

Theorem 4.

If Assumptions A and E hold and $s>\max \{ s_{0} + 2,d/2 + 4\}$ , then the solution of the effective equation exists for 0 ≤τ ≤ T, and for any s ₁ < s we have

$$\displaystyle{I(v^{\varepsilon }(\cdot ))\mathop{\longrightarrow}\limits_{\varepsilon \rightarrow 0}^{}I(\tilde{a}(\cdot ))\quad \text{in}\quad C([0,T],h_{I}^{s_{1} }).}$$

The proof of this theorem follows that of Theorem 1, with slight modifications. Cf. [8], where the result is proven for the non-resonant case.