New contiguity relation of the sixth Painlevé equation from a truncation

Details New contiguity relation of the sixth Painlevé equation from a truncation New contiguity relation of the sixth Painlevé equation from a truncation

2001-12-04

arxiv.org/abs/nlin/0112007v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

LaTex 2e. To appear, Physica D

Scientific paper

10.1016/S0167-2789(01)00372-4

For the master Painlev\'e equation P6(u), we define a consistent method, adapted from the Weiss truncation for partial differential equations, which allows us to obtain the first degree birational transformation of Okamoto. Two new features are implemented to achieve this result. The first one is the homography between the derivative of the solution $u$ and a Riccati pseudopotential. The second one is an improvement of a conjecture by Fokas and Ablowitz on the structure of this birational transformation. We then build the contiguity relation of P6, which yields one new second order nonautonomous discrete equation.

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