Reciprocal transformations and deformations of integrable hierarchies

TL;DR

This paper introduces variable transformations that convert newly discovered integrable hierarchies into well-known Harry-Dym and mKdV hierarchies, enhancing understanding of their structure and relationships.

Contribution

It provides explicit reciprocal transformations that relate recent integrable hierarchies to classical ones, clarifying their connections.

Findings

Transformations map new hierarchies to classical integrable systems

Reveals structural relationships between different hierarchies

Enhances understanding of integrable hierarchy deformations

Abstract

We present changes of variables that transform new integrable hierarchies found by Szablikowski and B{\l}aszak using the $R$-matrix deformation technique [J. Math. Phys. 47 (2006), paper 043505, nlin.SI/0501044] into known Harry-Dym-type and mKdV-type hierarchies.