# Liouville Correspondences between Integrable Hierarchies

## TL;DR

This paper explores explicit mathematical correspondences between different integrable hierarchies, revealing how transformations relate their spectral problems, hierarchies, and conservation laws, and establishing connections among multiple integrable equations.

## Contribution

It introduces explicit Liouville transformations linking Novikov, Sawada-Kotera, Degasperis-Procesi, and Kaup-Kupershmidt hierarchies, and constructs an implicit relationship among these equations.

## Key findings

- Liouville transformations relate isospectral problems of the hierarchies.
- Hierarchies are connected through positive and negative directions.
- An implicit relationship between Novikov and Degasperis-Procesi equations is established.

## Abstract

In this paper, we study explicit correspondences between the integrable Novikov and Sawada-Kotera hierarchies, and between the Degasperis-Procesi and Kaup-Kupershmidt hierarchies. We show how a pair of Liouville transformations between the isospectral problems of the Novikov and Sawada-Kotera equations, and the isospectral problems of the Degasperis-Procesi and Kaup-Kupershmidt equations relate the corresponding hierarchies, in both positive and negative directions, as well as their associated conservation laws. Combining these results with the Miura transformation relating the Sawada-Kotera and Kaup-Kupershmidt equations, we further construct an implicit relationship which associates the Novikov and Degasperis-Procesi equations.