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Time-Averaging for Weakly Nonlinear CGL Equations with Arbitrary Potentials

  • Guan Huang
  • Sergei KuksinEmail author
  • 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.

Keywords

Effective Parenting Random Force Integrable Nonlinear PDEs Unperturbed Linear System Constant Coefficient PDE 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.CMLS, Ecole PolytechniquePalaiseauFrance
  2. 2.IMJ, Université Paris Diderot-Paris 7Paris Cedex 13France
  3. 3.Université de Cergy-PontoiseCergy-PontoiseFrance

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