Solution of the KdV equation on the line with analytic initial potential

Abstract

We present a theory of Sturm-Liouville non-symmetric vessels, realizing an inverse scattering theory for the Sturm-Liouville operator with analytic potentials on the line. This construction is equivalent to the construction of a matrix spectral measure for the Sturm-Liouville operator, defined with an analytic potential on the line. Evolving such vessels we generate KdV vessels, realizing solutions of the KdV equation. As a consequence, we prove the following theorem: Suppose that q(x) is an analytic function on R. There exists a KdV vessel, which exists on a subset O of the plane. For each real x there exists positive T_x such that $\{x\}\times [-T_x,T_x]$ is in O. The potential q(x) is realized by the vessel for t=0. Since we also show that if q(x,t) is a solution of the KdV equation on a strip $R\times[0,T]$, then there exists a vessel, realizing it, the theory of vessels becomes a universal tool to study this problem. Finally, we notice that the idea of the proof applies to a similar existence of a solution for evolutionary NLS and Boussinesq equations, since both of these equations possess vessel constructions.