On the Korteweg–de Vries Equation: Convergent Birkhoff Normal Form

The Korteweg–de Vries equation (KdV)[formula]is a completely integrable Hamiltonian system of infinite dimension with phase space the Sobolev spaceHN(S1; R), (N⩾1), Hamiltonian H(q):=∫S1(12(∂xq(x))2+q(x)3) dx, and Poisson structure ∂/∂x. The functionq≡0 is an elliptic fixed point. We prove that for anyN⩾1, the Korteweg–de Vries equation (and thus the entire KdV-hierarchy) admits globally defined real analytic action-angle variables. As a consequence it follows that in a neighborhood ofq≡0 inH1(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 vectorfield of KdV.