Advertisement

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)

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.

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).

References

  1. 1.
    Arnold, V., Kozlov, V.V., Neistadt, A.I.: Mathematical Aspects of Classical and Celestial Mechanics, 3rd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  2. 2.
    Bambusi, D.: Galerkin averaging method and Poincaré normal form for some quasilnear PDEs. Ann. Scuola Norm. Sup. Pisa C1. Sci. 4, 669–702 (2005)Google Scholar
  3. 3.
    Bogoljubov, N.N., Mitropol’skij, J.A.: Asymptotic Methods in the Theory of Non-linear Oscillations. Gordon & Breach, New York (1961)Google Scholar
  4. 4.
    Dymov, A.: Nonequilibrium statistical mechanics of weakly stochastically perturbed system of oscillators. Preprint (2015) [arXiv: 1501.04238]Google Scholar
  5. 5.
    Faou, E., Germain, P., Hani, Z.: The weakly nonlinear large box limit of the 2D cubic nonlinear Schrödinger equation. Preprint (2013) [arXiv1308.6267]Google Scholar
  6. 6.
    Freidlin, M.I., Wentzell, A.D.: Averaging principle for stochastic perturbations of multifrequency systems. Stoch. Dyn. 3, 393–408 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Gérard, P., Grellier, S.: Effective integrable dynamics for a certain nonlinear wave equation. Anal. PDE 5, 1139–1154 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Huang, G.: An averaging theorem for nonlinear Schrödinger equations with small nonlinearities. DCDS-A 34(9), 3555–3574 (2014)zbMATHCrossRefGoogle Scholar
  9. 9.
    Huang, G.: Long-time dynamics of resonant weakly nonlinear CGL equations. J. Dyn. Diff. Equat. 1–13 (2014). doi: 10.1007/s10884-014-9391-0
  10. 10.
    Huang, G., Kuksin, S.B.: KdV equation under periodic boundary conditions and its perturbations. Nonlinearity 27, 1–28 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kelley, J.L., Namioka, I.: Linear Topological Spaces. Springer, New York/Heidelberg (1976)zbMATHGoogle Scholar
  12. 12.
    Khasminski, R.: On the averaging principle for Ito stochastic differential equations. Kybernetika 4, 260–279 (1968) (in Russian)MathSciNetGoogle Scholar
  13. 13.
    Kuksin, S.: Damped-driven KdV and effective equations for long-time behavior of its solutions. Geom. Funct. Anal. 20, 1431–1463 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Kuksin, S.: Weakly nonlinear stochastic CGL equations. Ann. Inst. H. Poincaré B 49(4), 1033–1056 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Kuksin, S., Maiocchi, A.: Resonant averaging for weakly nonlinear stochastic Schrödinger equations. Nonlinearity 28, 2319–2341 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kuksin, S., Maiocchi, A.: The limit of small Rossby numbers for randomly forced quasi-geostrophic equation on β-plane. Nonlinearity 28, 2319–2341 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kuksin, S., Piatnitski, A.: Khasminskii-Whitham averaging for randomly perturbed KdV equation. J. Math. Pures Appl. 89, 400–428 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Lochak, P., Meunier, C.: Multiphase Averaging for Classical Systems. Springer, New York/Berlin/Heidelberg (1988)zbMATHCrossRefGoogle Scholar
  19. 19.
    Nazarenko, S.: Wave Turbulence. Springer, Berlin (2011)zbMATHCrossRefGoogle Scholar
  20. 20.
    Runst, T., Sickel, W.: Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, vol. 3. de Gruyter, Berlin (1996)zbMATHCrossRefGoogle Scholar
  21. 21.
    Stroock, D., Varadhan, S.R.S.: Multidimensional Diffusion Processes. Springer, Berlin (1979)zbMATHGoogle Scholar
  22. 22.
    Zygmund, A.: Trigonometric Series, vol. II, 3th edn. Cambridge University Press, Cambridge (2002)Google Scholar

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

Personalised recommendations