A symmetric difference-differential Lax pair for Painlevé VI

2016/03/14 by Ormerod, Christopher M., Rains, Eric M.
#14H70 #39A10 #39A13 #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences

paper · doi:10.48550/arxiv.1603.04393

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.

Related

• Lax pairs for additive difference Painlevé equations
• Quivers and difference Painleve equations
• Towards second order Lax pairs to discrete Painlevé equations of first degree
• Lax formalism for q-Painleve equations with affine Weyl group symmetry of type E⁽¹⁾_n
• A new Lax pair for the sixth Painlevé equation associated with \mathfrakso(8)