Abstract
Consider weakly nonlinear complex Ginzburg–Landau (CGL) equation of the form:
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:
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
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- 1.
This is fulfilled, for example, if (i) holds and \(\mathcal{P}(u) = -u + \mathcal{P}_{0}(u)\), where the Lipschitz constant of \(\mathcal{P}_{0}\) is less than one.
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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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Appendix
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 \(\tau =\varepsilon t\):
where
and introduce the effective equation
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
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.
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Huang, G., Kuksin, S., Maiocchi, A. (2015). Time-Averaging for Weakly Nonlinear CGL Equations with Arbitrary Potentials. In: Guyenne, P., Nicholls, D., Sulem, C. (eds) Hamiltonian Partial Differential Equations and Applications. Fields Institute Communications, vol 75. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2950-4_11
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