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Time-Averaging for Weakly Nonlinear CGL Equations with Arbitrary Potentials

Abstract : Consider weakly nonlinear complex Ginzburg-Landau (CGL) equation of the form: u t C i. 4u C V.x/u/ D " u C "; u/; x 2 R d ; (*) under the periodic boundary conditions, where 0 and P is a smooth function. Let f 1 .x/; 2 .x/; : : : g be the L 2-basis formed by eigenfunctions of the operator 4 C V.x/. For a complex function u.x/, write it as u.x/ D P k 1 v k k .x/ and set I k .u/ D 1 2 jv k j 2. Then for any solution u.t; x/ of the linear equation. / "D0 we have I.u.t; // D 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 " 1 , then for any its solution u " .t; x/, the limiting behavior of the curve I.u " .t; // on time intervals of order " 1 , as " ! 0, can be uniquely characterized by a solution of a certain well-posed effective equation: u t D " 4u C "F.u/; where F.u/ is a resonant averaging of the nonlinearity; u/. We also prove similar results for the stochastically perturbed equation, when a white in time and
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Submitted on : Friday, August 28, 2020 - 10:34:06 PM
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Guan Huang, Sergei Kuksin, Alberto Maiocchi. Time-Averaging for Weakly Nonlinear CGL Equations with Arbitrary Potentials. Hamiltonian Partial Differential Equations and Applications, 75, pp.323-349, 2015, Fields Institute Communications, ⟨10.1007/978-1-4939-2950-4_11⟩. ⟨hal-02922517⟩



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