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Journal of Dynamical and Control Systems

, Volume 21, Issue 3, pp 379–400 | Cite as

On Longtime Dynamics of Perturbed KdV Equations

  • Guan Huang
Article
  • 145 Downloads

Abstract

Consider a perturbed KdV equation:
$$u_t+u_{xxx}-6uu_x=\epsilon f(u)(x),\quad x\in\mathbb{T}=\mathbb{R}/\mathbb{Z},\;\int_{\mathbb{T}}u(x,t)dx=0, $$
where the nonlinear perturbation defines analytic operators u(⋅)↦f(u(⋅)) in sufficiently smooth Sobolev spaces. Assume that the equation has an 𝜖-quasi-invariant measure μ and satisfies some additional mild assumptions. Let u 𝜖 (t) be a solution. Then on time intervals of order 𝜖 −1, as 𝜖→0, its actions I(u 𝜖 (t,⋅)) can be approximated by solutions of a certain well-posed averaged equation, provided that the initial datum is μ-typical.

Keywords

KdV Perturbations Longtime behaviour Averaging Gibbs measure 

Mathematics Subject Classifications (2010)

35Q53 70K65 34C29 37L40 74H40 

Notes

Acknowledgments

The author wants to thank his PhD supervisor professor Sergei Kuksin for formulation of the problem and guidance. He would also like to thank all of the staff and faculty at CMLS of École Polytechnique for their support.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.C.M.L.SÉcole PolytechniquePalaiseauFrance

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