Three theorems on perturbed KdV

  • Sergei B. Kuksin
Conference paper
Part of the NATO Science for Peace and Security Series book series (NAPSB)

This short paper is based on a lecture, given at the NATO Advanced Study Institute on Hamiltonian dynamical systems (Montréal, 2007). Its goal is to discuss three theorems on the long-time behaviour of solutions of a perturbed KdV equation under periodic boundary conditions. These theorems are infinite-dimensional analogies of three classical results on small perturbations of an integrable finite dimensional system:

  • The KAM theorem

  • The first-order averaging theory for Hamiltonian perturbations

  • The Khasminskii averaging theory for random perturbations

The three theorems raise many new questions, some of which are mentioned below.

We stress that the three theorems are infinite-dimensional analogies of some finite-dimensional statements. That is, for nearly integrable nonlinear PDEs (under periodic boundary conditions) we do not know any result which is essentially infinite-dimensional. There are no doubts that such results exist. To find them is a big challenge.


Normal Form Periodic Boundary Condition Random Perturbation Nondegeneracy Condition NATO Advance Study Institute 
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Copyright information

© Springer Science + Business Media B.V 2008

Authors and Affiliations

  • Sergei B. Kuksin
    • 1
  1. 1.Department of MathematicsHeriot-Watt UniversityUK

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