A symmetric difference-differential Lax pair for Painlev\'e VI

TL;DR

This paper introduces a novel symmetric difference-differential Lax pair for Painlevé VI, linking continuous and discrete isomonodromic deformations and deriving related Painlevé equations through degenerations.

Contribution

It presents a new Lax pair structure for Painlevé VI that unifies difference and differential equations with symmetry, extending to other Painlevé equations via degenerations.

Findings

Derived a symmetric difference-differential Lax pair for Painlevé VI.

Connected discrete isomonodromic deformations to a discrete Painlevé V.

Obtained Lax pairs for Painlevé V and degenerate Painlevé III equations.

Abstract

We present a Lax pair for the sixth Painlev\'e equation arising as a continuous isomonodromic deformation of a system of linear difference equations with an additional symmetry structure. We call this a symmetric difference-differential Lax pair. We show how the discrete isomonodromic deformations of the associated linear problem gives us a discrete version of the fifth Painlev\'e equation. By considering degenerations we obtain symmetric difference-differential Lax pairs for the fifth Painlev\'e equation and the various degenerate versions of the third Painlev\'e equation.