A Riemann--Hilbert approach to Painlevé IV

Abstract

This paper applies methods of Van der Put and Van derPut-Saito to the fourth Painlevé equation. One obtains a Riemann--Hilbert correspondence between moduli spaces of rank two connections on ${\mathbb{P}}^1$ and moduli spaces for the monodromy data. The moduli spaces for these connections are identified with Okamoto--Painlevé varieties and the Painlevé property follows. For an explicit computation of the full group of Bäcklund transformations, rank three connections on ${\mathbb{P}}^1$ are introduced, inspired by the symmetric form for $\mathrm{PIV}$ as was studied by M. Noumi and Y. Yamada.

arXiv comment18 pages

References in corpus (4)

• Moduli of Stable Parabolic Connections, Riemann-Hilbert correspondence and Geometry of Painlevé equation of type VI, Part I
• Painlevé equations and deformations of rational surfaces with rational double points
• Moduli of parabolic connections on a curve and Riemann-Hilbert correspondence
• Nodal curves and Riccati solutions of Painlevé equations

Cited by in corpus (2)

• Geometric Aspects of the Painlevé Equations $\mathrm{PIII}({\mathrm{D}}_6)$ and $\mathrm{PIII}({\mathrm{D}}_7)$
• Isomonodromy for the Degenerate Fifth Painlevé Equation