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

The Korteweg-de Vries equation (KdV) partial derivative(t) upsilon(x, t) + partial derivative(x)(3) upsilon(x, t) - 3 partial derivative(x) upsilon(x, t)(2) = 0 (x is an element of S-1,t is an element of R) is a completely integrable system with phase space L-2(S-1). although the Hamiltonian H(q) := integral(S1) (1/2(partial derivative(x)q(x))(2) + q(x)(3)) dx is defined only on the dense subspace H-1(S1), we prove that the frequencies omega(j) = partial derivative H/partial derivative J(j) can be defined on the whole space L-2(S-1), where (J(j))(j greater than or equal to 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.