Rational Solutions of Painlev\'e V from Hankel Determinants and the Asymptotics of Their Pole Locations

TL;DR

This paper investigates the asymptotic distribution of poles of rational solutions to the fifth Painlevé equation by analyzing Hankel determinants and their roots, revealing a region filled with roots bounded by analytic arcs.

Contribution

It introduces a novel approach linking tau functions to Hankel determinants to study pole distributions of Painlevé V solutions, extending previous methods used for Painlevé II.

Findings

Roots asymptotically fill a region bounded by analytic arcs

Location of roots can be approximated by quantization conditions

Provides insight into pole distribution for large degree solutions

Abstract

In this paper, we analyze the asymptotic behaviour of the poles of certain rational solutions of the fifth Painlev\'e equation. These solutions are constructed by relating the corresponding tau function to a Hankel determinant of a certain sequence of moments. This approach was also used by one of the authors and collaborators in the study of the rational solutions of the second Painlev\'e equation. More specifically, we study the roots of the corresponding polynomial tau function, whose location corresponds to the poles of the associated rational solution. We show that, upon suitable rescaling, the roots asymptotically fill a region bounded by analytic arcs when the degree of the polynomial tau function tends to infinity and the other parameters are kept fixed. Moreover, we provide an approximate location of these roots within the region in terms of suitable quantization conditions.