A generalised multicomponent system of Camassa-Holm-Novikov equations

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.