Miura-reciprocal transformations for non-isospectral Camassa-Holm hierarchies in $2+1$ dimensions

TL;DR

This paper introduces two related hierarchies of PDEs in 2+1 dimensions connected through reciprocal and Miura transformations, linking them to the Calogero-Bogoyavlenski-Schiff equation and its modified form.

Contribution

It demonstrates how reciprocal and Miura transformations relate two hierarchies of PDEs in 2+1 dimensions, establishing a modified relationship between them.

Findings

Hierarchies connected via reciprocal transformations

One hierarchy is a modified version of the other

Connections to Calogero-Bogoyavlenski-Schiff equation

Abstract

We present two hierarchies of partial differential equations in $2+1$ dimensions. Since there exist reciprocal transformations that connect these hierarchies to the Calogero-Bogoyavlenski-Schiff equation and its modified version, we can prove that one of the hierarchies can be considered as a modified version of the other. The connection between them can be achieved by means of a combination of reciprocal and Miura transformations.