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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.

Keywords

Normal Form Periodic Boundary Condition Random Perturbation Nondegeneracy Condition NATO Advance Study Institute 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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