The Quasi-Integrability of a Generalized Camassa-Holm Equation

TL;DR

This paper investigates a generalized Camassa-Holm equation's integrability, revealing that it retains a bi-Hamiltonian structure only in its original form, thus highlighting the limits of its quasi-integrability.

Contribution

It introduces a framework to analyze the quasi-integrability of a generalized Camassa-Holm equation and shows that the bi-Hamiltonian structure exists only for the original equation.

Findings

Bi-Hamiltonian structure exists only for the original Camassa-Holm equation.

The generalized equation is not quasi-integrable unless it reduces to the original form.

The framework of Dubrovin's bi-Hamiltonian deformations is effective for analyzing integrability.

Abstract

This paper examines a generalization of the Camassa-Holm equation from the perspective of integrability. Using the framework developed by Dubrovin on bi-Hamiltonian deformations and the general theory of quasi-integrability, we demonstrate that a unique bi-Hamiltonian structure is possible for this generalized equation only when it reduces to the original CH equation.