Special solutions to five autonomous integrable partial difference equations via the third and sixth Painlev\'e equations and the Garnier system in two variables

TL;DR

This paper explores special solutions of five autonomous integrable partial difference equations, linking them to Painlevé equations and the Garnier system through Bäcklund transformations.

Contribution

It demonstrates that these PΔEs have special solutions described by non-autonomous difference equations derived from Painlevé and Garnier systems.

Findings

Special solutions are characterized by non-autonomous difference equations.

Connections established between autonomous PΔEs and Painlevé-type dynamics.

Provides new insights into the structure of integrable partial difference equations.

Abstract

In this paper, we study special solutions of five autonomous integrable partial difference equations (P$\Delta$Es). More precisely, we show that these P$\Delta$Es admit special solutions that are described by non-autonomous ordinary difference equations arising from B\"acklund transformations of the third and sixth Painlev\'e equations and the Garnier system in two variables. This result provides a new perspective on the relationship between autonomous integrable P$\Delta$Es and Painlev\'e-type dynamics.