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.

18 pages