Generalisations of the Camassa-Holm equation

TL;DR

This paper classifies generalized Camassa-Holm equations with infinite symmetries, linking them to integrable hierarchies, and provides Lax pairs or linearizations, revealing some potentially new equations.

Contribution

It introduces a classification of generalized Camassa-Holm equations with higher symmetries and connects them to integrable hierarchies, including new equations.

Findings

Identified classes of equations with infinite hierarchies of symmetries

Established Lax representations or linearizations for these equations

Discovered some potentially new integrable equations

Abstract

We classify generalised Camassa-Holm type equations which possess infinite hierarchies of higher symmetries. We show that the obtained equations can be treated as negative flows of integrable quasi-linear scalar evolution equations of orders 2, 3 and 5. We present the corresponding Lax representations or linearisation transformations for these equations. Some of the obtained equations seem to be new.