Meromorphic Painlev\'e III transcendents and the Joukowski correspondence

TL;DR

This paper explores a geometric approach to Painlevé III transcendents via a twistor correspondence derived from the Joukowski map, revealing conditions for meromorphic solutions and quantization of parameters.

Contribution

It introduces a novel geometric framework linking the Joukowski map to Painlevé III solutions and establishes a quantization condition based on monodromy data.

Findings

Solutions are meromorphic at the origin under axial simplicity.

Quantization condition for the Painlevé III parameter is derived.

Trivial Stokes matrices correspond to specific monodromy data.

Abstract

We study a twistor correspondence based on the Joukowski map reduced from one for stationary-axisymmetric self-dual Yang-Mills and adapt it to the Painlev\'e III equation. A natural condition on the geometry (axissimplicity) leads to solutions that are meromorphic at the fixed singularity at the origin. We show that it also implies a quantisation condition for the parameter in the equation. From the point of view of generalized monodromy data, the condition is equivalent to triviality of the Stokes matrices and half-integral exponents of formal monodromy. We obtain canonically defined representations in terms of a Birkhoff factorization whose entries are related to the data at the origin and the Painlev\'e constants.