A symmetric difference-differential Lax pair for Painlevé VI

Creators: Ormerod, Christopher M.; Rains, Eric

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Abstract

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.

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© 2017 The authors. Published by Oxford University Press. This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited. Received on 24 December 2016; editorial decision on 26 February 2017; accepted on 26 February 2017; Published: 08 May 2017. We would like to thank Frank Nijhoff for his correspondence and bringing our attention to [22], ER was supported by a grant from the National Science Foundation, [DMS-1500806]. We would also like to thank the anonymous referees for their diligence.

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