A Riemann--Hilbert approach to Painlev\'e IV

TL;DR

This paper develops a Riemann-Hilbert framework for Painlevé IV, linking moduli spaces of connections and monodromy data, and explores Bäcklund transformations via rank three connections.

Contribution

It introduces a Riemann-Hilbert correspondence for Painlevé IV and computes the Bäcklund transformation group using rank three connections.

Findings

Established a Riemann-Hilbert correspondence for PIV

Identified moduli spaces with Okamoto–Painlevé varieties

Computed the full Bäcklund transformation group

Abstract

This paper applies methods of Van der Put and Van derPut-Saito to the fourth Painlev\'e 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--Painlev\'e varieties and the Painlev\'e property follows. For an explicit computation of the full group of B\"acklund transformations, rank three connections on $\mathbb{P}^1$ are introduced, inspired by the symmetric form for ${\rm PIV}$ as was studied by M. Noumi and Y. Yamada.