Three theorems on perturbed KdV
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.
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