On existence and properties of roots of third Painlev\'e' transcendents

TL;DR

This paper investigates the roots of third Painlevé transcendents, establishing their existence, properties, and providing an efficient algorithm for approximating solutions near roots and poles, with numerical validation and bounds on root-pole distances.

Contribution

It introduces a novel approach using Hamiltonian formalism and integral equations to analyze roots of P_III-functions, including an algorithm for approximate solutions and bounds on root-pole distances.

Findings

Existence of roots at non-zero points established.

An efficient power series algorithm for root approximation developed.

Bounds on distances between roots and poles derived.

Abstract

Separate consideration of properties of roots of Third Painlev\'e transcendents (P_III-functions) is necessary due to irregularity the differential equation defining them reveals on the subset of the phase space where its solution would vanish. Application of the Hamiltonian formalism enables one to replace the mentioned second order differential equation (Third Painlev\'e equation) by two independent systems of two nonlinear first order equations whose structures allow to name them coupled Riccati equations. The existence of P_III-functions vanishing at a given non-zero point then follows, all they being analytic thereat. The set $\mathbb{Z}_2\times \mathbb{C}$ (or $\mathbb{Z}_2\times \mathbb{R}$) can be used for their indexing. It proves also to be natural to use as an unknown the third order derivative rather than the original nknown itself. After transformation of the corresponding differential equations to equivalent integral equations the efficient algorithm of the constructing of approximate solutions to Third Painlev\'e equation in vicinity of their non-zero root in the form of truncated power series is obtained. An example of its application is given, its numerical validation presenting results in a graphical form is carried out. The associated approximation applicable in vicinity of a pole of the corresponding P_III-function is given as well. The bounds from below for the distances between a pair of roots of a P_III-function and between a root and a pole representable in terms of elementary functions are derived.