A generalised multicomponent system of Camassa-Holm-Novikov equations

TL;DR

This paper introduces a generalized two-component system that extends the Camassa-Holm and Novikov equations, analyzing its symmetries, conservation laws, and solutions, revealing special properties at specific parameter values.

Contribution

It presents a unified framework for the Camassa-Holm and Novikov equations, including symmetry analysis, conservation laws, and peakon solutions, especially highlighting the case when b=2.

Findings

For b≠2, the system has a 3-dimensional symmetry algebra.

For b=2, the system has a 6-dimensional symmetry algebra and higher symmetries.

The system admits peakon solutions, including non-constant amplitude 1-peakons at b=2.

Abstract

In this paper we introduce a two-component system, depending on a parameter $b$, which generalises the Camassa-Holm ($b=1$) and Novikov equations ($b=2$). By investigating its Lie algebra of classical and higher symmetries up to order $3$, we found that for $b\neq 2$ the system admits a $3$-dimensional algebra of point symmetries and apparently no higher symmetries, whereas for $b=2$ it has a $6$-dimensional algebra of point symmetries and also higher order symmetries. Also we provide all conservation laws, with first order characteristics, which are admitted by the system for $b=1,2$. In addition, for $b=2$, we show that the system is a particular instance of a more general system which admits an $\mathfrak{sl}(3,\mathbb{R})$-valued zero-curvature representation. Finally, we found that the system admits peakon solutions and, in particular, for $b=2$ there exist 1-peakon solutions with non-constant amplitude.