Journal of Dynamical and Control Systems

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

On Longtime Dynamics of Perturbed KdV Equations

  • Guan Huang


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.


KdV Perturbations Longtime behaviour Averaging Gibbs measure 

Mathematics Subject Classifications (2010)

35Q53 70K65 34C29 37L40 74H40 



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.


  1. 1.
    Aron R, Cima J. A theorem on holomorphic mappings into Banach spaces with basis. Proc J Am Math Soc 1972;36(1).Google Scholar
  2. 2.
    Bogachev V. Differentiable measures and the Malliavin calculus: American Mathematical Society; 2010.Google Scholar
  3. 3.
    Bogachev V, Malofeev I. On the absolute continuity of the distributions of smooth functions on infinite-dimensional spaces with measures. preprint; 2013.Google Scholar
  4. 4.
    Bourgain J. Global solutions of nonlinear Schrödinger equations. vol. 46. American Mathematical Society; 1999.Google Scholar
  5. 5.
    Burq N, Tzvetkov N. Random data cauchy theory for supercritical wave equations ii: a global existence result. Invent Math 2008;173(3):477–96.CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Dudley R. Real analysis and probability. Cambridge University Press; 2002.Google Scholar
  7. 7.
    Huang G. An averaging theorem for a perturbed KdV equation. Nonlinearity 2013; 26(6):1599.CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Huang G, Kuksin S. KdV equation under periodic boundary conditions and its perturbations. preprint, 2013. arXiv: 1309.1597.
  9. 9.
    Kappeler T, Pöschel J. KAM & KdV. Springer; 2003.Google Scholar
  10. 10.
    Kappeler T, Schaad B, Topalov P. Qualtitative features of periodic solutions of KdV. Commun Partial Diff Equat. 2013. arXiv: 1110.0455.
  11. 11.
    Kuksin S. Analysis of Hamiltonian PDEs. Oxford: Oxford University Press; 2000.Google Scholar
  12. 12.
    Kuksin S, Perelman G. Vey theorem in infinite dimensions and its application to KdV. DCDS-A 2010;27:1–24.CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Kuksin S, Piatnitski A. Khasminskii-whitham averaging for randomly perturbed KdV equation. J Math Pures Appl 2008;89:400–428.CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Lochak P, Meunier C. Multiphase averaging for classical systems (with applications to adiabatic theorems). Springer; 1988.Google Scholar
  15. 15.
    Zhidkov P. Korteweg-de Vries and nonlinear Schrödinger equations: qualitative theory. vol. 1756. Springer Verlag; 2001.Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

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

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