Abstract from the Pacific Journal of Mathematics

Volume 181, Number 1, November 1997

Title: On the Korteweg-de Vries Equation: Frequencies and Initial Value Problem

Author: D. Baettig, T. Kappeler and B. Mityagin

Abstract:

The Korteweg-de Vries equation (KdV)

\partial_t v(x,t) + \partial_x^3 v(x, t) - 3 \partial_x v(x,t)^2 = 0
(x \in S^1, t \in R)

is a completely integrable system with phase space L^2(S^1). Although the Hamiltonian

H(q) := \int_{S^1}(1/2 (\partial_x q(x))^2 + q(x)^3)dx

is defined only on the dense subspace H^1(S^1), we prove that the frequencies \omega_j = \frac {\partial H}{\partial J_j} can be defined on the whole space L^2(S^1), where (J_j)_{j \ge 1} denote the action variables which are globally defined on L^2(S^1). These frequencies are real analytic functionals and can be used to analyze Bourgain's weak solutions of KdV with initial data in L^2(S^1). The same method can be used for any equation in the KdV-hierarchy.

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