On the Korteweg–de Vries equation: Frequencies and initial value problem

D. Bättig, T. Kappeler and B. Mityagin

Vol. 181 (1997), No. 1, 1–55

DOI: 10.2140/pjm.1997.181.1

Abstract

The Korteweg-de Vries equation (KdV)

is a completely integrable system with phase space L²(S¹). Although the Hamiltonian ℋ(q) := ∫ _S¹dx is defined only on the dense subspace H¹(S¹), we prove that the frequencies ω_j = can be defined on the whole space L²(S¹), where (J_j)_j≥1 denote the action variables which are globally defined on L²(S¹). These frequencies are real analytic functionals and can be used to analyze Bourgain’s weak solutions of KdV with initial data in L²(S¹). The same method can be used for any equation in the KdV −hierarchy.

Milestones

Received: 26 February 1996

Published: 1 November 1997