On the Korteweg-de Vries equation: frequencies and initial value problem

Unspecified Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: http://doi.org/10.5167/uzh-22229 Originally published at: Bättig, D; Kappeler, T; Mityagin, B (1997). On the Korteweg-de Vries equation: frequencies and initial value problem. Pacific Journal of Mathematics, 181(1):1-55. pacific journal of mathematics Vol. 181, No. 1, 1997 ON THE KORTEWEG-DE VRIES EQUATION: FREQUENCIES AND INITIAL VALUE PROBLEM D. Bättig, T. Kappeler and B. Mityagin The Korteweg-de Vries equation (KdV) ∂tv(x, t) + ∂ xv(x, t)− 3∂xv(x, t) = 0 (x ∈ S, t ∈ R) is a completely integrable system with phase space L(S). Although the Hamiltonian H(q) := ∫ S1 ( 1 2 ( ∂xq(x) )2 + q(x)3) dx is defined only on the dense subspace H(S), we prove that the frequencies ωj = ∂H ∂Jj can be defined on the whole space L(S), where (Jj)j≥1 denote the action variables which are globally defined on L(S). These frequencies are real analytic functionals and can be used to analyze Bourgain’s weak solutions of KdV with initial data in L(S). The same method can be used for any equation in the KdV−hierarchy.