Lax pairs for additive difference Painlev\'e equations

TL;DR

This paper explicitly constructs Lax pairs for additive difference Painlevé equations of various types, providing a new tool for analyzing these integrable systems and their compatibility conditions.

Contribution

The paper introduces explicit scalar Lax pairs for additive difference Painlevé equations of types E7, E6, D4, and A3, including degenerations, advancing the understanding of their integrability.

Findings

Explicit Lax pair for E7^{(1)} type Painlevé equation.

Degenerated Lax pairs for E6^{(1)}, D4^{(1)}, and A3^{(1)} types.

Proof of compatibility using coefficient characterization.

Abstract

A Lax pair for the additive difference Painlev\'e equation of type $E_7^{(1)}$ is explicitly obtained as certain linear difference equations of scalar form. The compatibility of the Lax pair is proved by using certain characterization of the coefficients in the Lax equation. Some Lax pairs for types $E_6^{(1)}$, $D_4^{(1)}$ and $A_3^{(1)}$ are also given by the degeneration.