Rational Solutions of the Painlev\'e-III Equation

TL;DR

This paper develops a Riemann-Hilbert approach to analyze the asymptotic behavior of rational solutions of the Painlevé-III equation, especially as the parameter n becomes large, revealing patterns in pole-zero distributions.

Contribution

It introduces a new Riemann-Hilbert representation for Painlevé-III rational solutions, enabling asymptotic analysis and explicit finite-dimensional systems for large n.

Findings

Distribution patterns of poles and zeros as n increases

Riemann-Hilbert representation for rational solutions

Finite-dimensional Hankel system for large n

Abstract

All of the six Painlev\'e equations except the first have families of rational solutions, which are frequently important in applications. The third Painlev\'e equation in generic form depends on two parameters $m$ and $n$, and it has rational solutions if and only if at least one of the parameters is an integer. We use known algebraic representations of the solutions to study numerically how the distributions of poles and zeros behave as $n\in\mathbb{Z}$ increases and how the patterns vary with $m\in\mathbb{C}$. This study suggests that it is reasonable to consider the rational solutions in the limit of large $n\in\mathbb{Z}$ with $m\in\mathbb{C}$ being an auxiliary parameter. To analyze the rational solutions in this limit, algebraic techniques need to be supplemented by analytical ones, and the main new contribution of this paper is to develop a Riemann-Hilbert representation of the rational solutions of Painlev\'e-III that is amenable to asymptotic analysis. Assuming further that $m$ is a half-integer, we derive from the Riemann-Hilbert representation a finite dimensional Hankel system for the rational solution in which $n\in\mathbb{Z}$ appears as an explicit parameter.