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arXiv · 1603.04393

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

Christopher M. Ormerod·Eric M. Rains
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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.

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Christopher M. Ormerod, Eric M. Rains. 2016-03-14. A symmetric difference-differential Lax pair for Painlev\'e VI. https://arxiv.org/abs/1603.04393

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