On isospectral deformations of an inhomogeneous string

TL;DR

This paper explores a broad class of isospectral deformations of inhomogeneous strings, generalizing known deformations related to the Camassa-Holm equation, and demonstrates their solvability via inverse spectral methods, including cases with discrete measures.

Contribution

It introduces new isospectral deformation equations for inhomogeneous strings that depend on boundary conditions and shows they can be linearized and solved using inverse spectral techniques.

Findings

Deformations lead to evolution equations on mass density.

Equations linearize on the spectral side.

Solvable with inverse spectral method, including discrete measures.

Abstract

In this paper we consider a class of isospectral deformations of the inhomogeneous string boundary value problem. The deformations considered are generalizations of the isospectral deformation that has arisen in connection with the Camassa-Holm equation for the shallow water waves. It is proved that these new isospectral deformations result in evolution equations on the mass density whose form depends on how the string is tied at the endpoints. Moreover, it is shown that the evolution equations in this class linearize on the spectral side and hence can be solved by the inverse spectral method. In particular, the problem involving a mass density given by a discrete finite measure and arbitrary boundary conditions is shown to be solvable by Stieltjes' continued fractions.