# Vibrations of an elastic bar, isospectral deformations, and modified Camassa-Holm equations

## TL;DR

This paper explores isospectral deformations of elastic rod vibrations modeled by Sturm-Liouville systems, linking them to a two-component modified Camassa-Holm equation and demonstrating its integrability and explicit solutions for discrete cases.

## Contribution

It introduces a unified approach to isospectral deformations related to the 2-mCH equation, extending previous work and analyzing the system's Hamiltonian structure and integrability.

## Key findings

- Complete solution for discrete multipeakon dynamics
- Proof of Liouville integrability of the system
- Unified framework for 1-mCH and 2-mCH equations

## Abstract

Vibrations of an elastic rod are described by a Sturm-Liouville system. We present a general discussion of isospectral (spectrum preserving) deformations of such a system. We interpret one family of such deformations in terms of a two-component modified Camassa-Holm equation (2-mCH) and solve completely its dynamics for the case of discrete measures (multipeakons). We show that the underlying system is Hamiltonian and prove its Liouville integrability. The present paper generalizes our previous work on interlacing multipeakons of the 2-mCH and multipeakons of the 1-mCH. We give a unified approach to both equations, emphasizing certain natural family of interpolation problems germane to the solution of the inverse problem for 2-mCH as well as to this type of a Sturm-Liouville system with singular coefficients.