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A symmetric difference-differential Lax pair for Painlevé VI

We present a Lax pair for the sixth Painlevé 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 Painlevé equation. By considering degenerations we obtain symmetric difference-differential Lax pairs for the fifth Painlevé equation and the various degenerate versions of the third Painlevé equation.

preprint2016arXivOpen access
Christopher M. OrmerodEric M. Rains
nlin.SI
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Christopher M. OrmerodEric M. Rains

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