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

Ontology type: schema:Chapter      Open Access: True

### Chapter Info

DATE

2015

AUTHORS

Guan Huang , Sergei Kuksin , Alberto Maiocchi

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. More... »

PAGES

323-349

### Book

TITLE

Hamiltonian Partial Differential Equations and Applications

ISBN

978-1-4939-2949-8
978-1-4939-2950-4

### Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/978-1-4939-2950-4_11

DOI

http://dx.doi.org/10.1007/978-1-4939-2950-4_11

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1036158147

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