ON THE KORTEWEG-DE VRIES EQUATION - CONVERGENT BIRKHOFF NORMAL-FORM

The Korteweg-de Vries equation (KdV) partial derivative(t)V(x,t)+parti al derivative(x)(3)V(x,t)-3 partial derivative(x)V(x,t)(2)=0 is a comp letely integrable Hamiltonian system of infinite dimension with phase space the Sobolev space H-N(S-1; R), (N greater than or equal to 1), H amiltonian H(q) := integral(S1)(1/2(partial derivative(x)q(x))(2) +q(x )(3)dx, and Poisson structure partial derivative/partial derivative x. The function q=D is an elliptic fixed point. We prove that for any N greater than or equal to 1, the Korteweg-de Vries equation (and thus t he entire KdV-hierarchy) admits globally defined real analytic action- angle variables. As a consequence it follows that in a neighborhood of q=O in H-1(S-1; R), the KdV-Hamiltonian H(and similarly any Hamiltoni an in the KdV-hierarchy) admits a convergent Birkhoff normal form; to the best of our knowledge this is the first such example in infinite d imension. Moreover, using the constructed action-angle variables, we a nalyze the regularity properties of the Hamiltonian vectorfield of KdV . (C) 1996 Academic Press, Inc.