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

Abstract

We present a consistent truncation, allowing us to obtain the first degree birational transformation found by Okamoto for the sixth Painlevé 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 Painlevé equations, ranging them in two distinct sequences.

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

References in corpus (2)

• New contiguity relation of the sixth Painlevé equation from a truncation
• On the Schlesinger transformations of the sixth Painlevé equation

Cited by in corpus (4)

• Exact solutions of nonlinear partial differential equations by singularity analysis
• Bäcklund Transformations of the Sixth Painlevé Equation in Terms of Riemann-Hilbert Correspondence
• New contiguity relation of the sixth Painlevé equation from a truncation
• A truncation for obtaining all the first degree birational transformations of the Painlevé transcendents