B\"acklund Transformations for First and Second Painlev\'e Hierarchies

TL;DR

This paper develops Bäcklund transformations for the entire first and second Painlevé hierarchies, extending known transformations and linking these hierarchies to new Painlevé-type equations.

Contribution

It introduces generalized Bäcklund transformations for Painlevé hierarchies, connecting them to novel Painlevé-type equations.

Findings

Generalized Bäcklund transformations for Painlevé hierarchies

Connections to new Painlevé-type equations

Extension of known transformations

Abstract

We give B\"acklund transformations for first and second Painlev\'e hierarchies. These B\"acklund transformations are generalization of known B\"acklund transformations of the first and second Painlev\'e equations and they relate the considered hierarchies to new hierarchies of Painlev\'e-type equations.