Time-Averaging for Weakly Nonlinear CGL Equations with Arbitrary Potentials

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


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.


Effective Parenting Random Force Integrable Nonlinear PDEs Unperturbed Linear System Constant Coefficient PDE 
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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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© 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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