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

TL;DR

This paper develops a vessel-based inverse scattering theory for the Sturm-Liouville operator with analytic potentials, enabling the construction of solutions to the KdV equation and related PDEs on the line.

Contribution

It introduces a novel vessel framework for solving the KdV equation with analytic initial potentials, extending to other integrable PDEs like NLS and Boussinesq.

Findings

Existence of KdV vessels for analytic potentials on the line.

Construction of solutions to the KdV equation using vessel theory.

Applicability of vessel methods to NLS and Boussinesq equations.

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.