Inverse problems associated with integrable equations of Camassa-Holm type; explicit formulas on the real axis, I

TL;DR

This paper revisits the inverse problem for the Camassa-Holm equation with discrete densities, deriving explicit formulas using orthogonal polynomials directly on the real axis, avoiding traditional string boundary transformations.

Contribution

It introduces a novel approach to solving the inverse problem for the Camassa-Holm equation using orthogonal polynomials on the real axis, simplifying previous methods.

Findings

Explicit formulas derived directly on the real axis

Method avoids transformation to string boundary problems

Applicable to discrete densities in Camassa-Holm equation

Abstract

The inverse problem which arises in the Camassa--Holm equation is revisited for the class of discrete densities. The method of solution relies on the use of orthogonal polynomials. The explicit formulas are obtained directly from the analysis on the real axis without any additional transformation to a "string" type boundary value problem known from prior works.