Miura Reciprocal transformation for two integrable hierarchies in 1+1 dimensions

TL;DR

This paper introduces two integrable PDE hierarchies in 1+1 dimensions related to shallow water wave equations, connected via reciprocal and Miura transformations to well-known integrable systems, revealing their deep structural relations.

Contribution

It constructs new hierarchies linked to shallow water equations and establishes their relation to CBS and mCBS equations through reciprocal and Miura transformations.

Findings

Hierarchies related to Camassa-Holm and Qiao equations are transformed into CBS and mCBS.

A relation between the initial hierarchies is derived via combined reciprocal and Miura transforms.

The work reveals structural links between shallow water equations and classical integrable systems.

Abstract

We here present two different hierarchies of PDEs in 1+1 dimensions whose first and second member are the shallow water wave Camassa-Holm and Qiao equations, correspondingly. These two hierarchies can be transformed by reciprocal methods into the Calogero-Bogoyanlevski-Schiff equation (CBS) and its modified version (mCBS), respectively. Considering that there exists a Miura transformation between the CBS and mCBS, we shall obtain a relation between the initial hierarchies by means of a composition of a Miura and reciprocal transforms.