Abstract

The Korteweg-de Vries (KdV) equation ∂tV(x,t)+∂x³V(x,t)−3∂xV(x,t)2=0 (x ∈ S1, t ∈ R) is a completely integrable Hamiltonian system of infinite dimension with phase space the Sobolev space HN(S1;R) (N≥1), Hamiltonian H(q):=∫S1(½(∂xq(x))2 + q(x)3)dx, and Poisson structure ∂/∂x. The function q≡0 is an elliptic fixed point. We prove that for any N≥1, the KdV equation (and thus the entire KdV hierarchy) admits globally defined real-analytic action-angle variables. As a consequence it follows that in a neighborhood of q≡0 in H1(S1;R), the KdV Hamiltonian H (and similarly any Hamiltonian in the KdV hierarchy) admits a convergent Birkhoff normal form; to the best of our knowledge this is the first such example in infinite dimension. Moreover, using the constructed action-angle variables, we analyze the regularity properties of the Hamiltonian vector field of the KdV equation.