On Transformations in the Painlev\'e Family

TL;DR

This paper proves that generic Painlevé equations from different families are orthogonal, indicating no general Bäcklund transformations exist between them, which clarifies the structural independence of these equations.

Contribution

It establishes the orthogonality of Painlevé equations across different families, a novel result in the understanding of their interrelations.

Findings

Painlevé equations from different families are orthogonal.

No general Bäcklund transformations exist between different Painlevé families.

This result clarifies the independence of Painlevé equations across families.

Abstract

In this paper we show that generic Painlev\'e equations from different families are orthogonal. In particular, this means that there are no general Backlund transformations between Painlev\'e equations from the different families $P_I-P_{VI}$ .