A Riemann–Hilbert approach to Painlevé IV

Abstract

The methods of [vdP-Sa, vdP1, vdP2] are applied to the fourth Painlevé equation. One obtains a Riemann–Hilbert correspondence between moduli spaces of rank two connections on ℙ¹ 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 ℙ¹ are introduced, inspired by the symmetric form for PIV, studied by M. Noumi and Y. Yamada.

Keywords:

• Moduli space for linear connections
• Irregular singularities
• Stokes matrices
• Monodromy spaces
• Isomonodromic deformations
• Painlevé equations

2000 Mathematics Subject Classification:

• 14D20
• 14D22
• 34M55