Resumen de Birkhoff coordinates for KdV on phase spaces of distributions

T. Kappeler, C. Möhr, P. Topalov

• The purpose of this paper is to extend the construction of Birkhoff coordinates for the KdV equation from the phase space of square integrable 1-periodic functions with mean value zero to the phase space H−10(T) of mean value zero distributions from the Sobolev space H−1(T) endowed with the symplectic structure (∂/∂x)−1. More precisely, we construct a globally defined real-analytic symplectomorphism Ω:H−10(T)→h−1/2 where h−1/2 is a weighted Hilbert space of sequences (xn,yn)n⩾1 supplied with the canonical Poisson structure so that the KdV Hamiltonian for potentials in H10(T) is a function of the actions ((x2n+y2n)/2)n⩾1 alone.