Non-isospectral 1+1 hierarchies arising from a Camassa--Holm hierarchy in 2+1 dimensions

TL;DR

This paper explores a non-isospectral Lax pair linked to a 2+1 dimensional hierarchy generalizing the Camassa-Holm equation, analyzing its symmetries and reductions to 1+1 dimensions.

Contribution

It introduces a non-isospectral hierarchy in 2+1 dimensions, investigates its non-classical Lie symmetries, and derives similarity reductions with non-isospectral 1+1 hierarchies.

Findings

Identified five arbitrary constants and three arbitrary functions in the symmetries.

Derived various similarity reductions of the hierarchy.

Found reduced hierarchies with non-isospectral 1+1 Lax pairs.

Abstract

The non-isospectral problem (Lax pair) associated with a hierarchy in 2+1 dimensions that generalizes the well known Camassa-Holm hierarchy is presented. Here, we have investigated the non-classical Lie symmetries of this Lax pair when the spectral parameter is considered as a field. These symmetries can be written in terms of five arbitrary constants and three arbitrary functions. Different similarity reductions associated with these symmetries have been derived. Of particular interest are the reduced hierarchies whose $1+1$ Lax pair is also non-isospectral.