Construction of a Sturm-Liouville vessel using Gelfand-Levitan theory. On solution of the Korteweg-de Vries equation in the first quadrant

TL;DR

This paper constructs a mathematical vessel using Gelfand-Levitan theory to solve the Korteweg-de Vries equation on the half line, providing new formulas, generalizations, and inverse scattering insights for potentials with singularities.

Contribution

It introduces a vessel construction method for potentials satisfying certain spectral regularity, enabling solutions to the KdV equation and generalizing inverse scattering theory.

Findings

Constructed a vessel for spectral functions with regularity conditions.

Provided formulas for Gelfand-Levitan equation components.

Extended the vessel theory to solve various equations like NLS and canonical systems.

Abstract

Using Gelfand-Levitan theory on a half line, we construct a vessel for the class of potentials, whose spectral functions satisfy a certain regularity assumption. When the singular part of the spectral measure is absent, we construct a canonical model of the vessel. Finally, evolving the constructed vessel, we solve the Korteweg de Vries equation on the half line, coinciding with the given potential for $t=0$. It is shown that the initial value for x=0 is prescribed by this construction, but can be perturbed using an "orthogonal" to the problem measure. The results, presented in this work 1. include formulas for the ingredients of the Gelfand-Levitan equation, 2. are shown to be general in the sense that NLS, Canonical systems and many more equations can be solved using theory of vessels, analogously to Zacharov-Shabath scheme, 3. present a generalized inverse scattering theory on a line for potentials with singularities using pre-vessels, 4. present the tau function and its role.