First degree birational transformations of the Painlevé equations and their contiguity relations

Details First degree birational transformations of the Painlevé equations and their contiguity relations First degree birational transformations of the Painlevé equations and their contiguity relations

2001-10-16

arxiv.org/abs/nlin/0110028v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

LaTex 2e. To appear, J. Phys. A, Special issue SIDE IV

Scientific paper

10.1088/0305-4470/34/48/315

We present a consistent truncation, allowing us to obtain the first degree birational transformation found by Okamoto for the sixth Painlev\'e equation. The discrete equation arising from its contiguity relation is then just the sum of six simple poles. An algebraic solution is presented, which is equivalent to but simpler than the Umemura solution. Finally, the well known confluence provides a unified picture of all first degree birational transformations for the lower Painlev\'e equations, ranging them in two distinct sequences.

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