Asymptotic behaviour of the third Painlev\'e transcendents in the space of initial values

TL;DR

This paper analyzes the asymptotic behavior of solutions to the third Painlevé equation, revealing properties of their limit sets and the distribution of poles and zeros near essential singularities.

Contribution

It provides a detailed description of the asymptotic behavior and pole-zero distribution of third Painlevé transcendents in the space of initial values.

Findings

Limit sets of solutions are compact and connected.

Solutions with essential singularities have infinitely many poles and zeros.

Behavior near infinity and zero is characterized in the space of initial values.

Abstract

We study the asymptotic behaviour of the solutions of the generic ($D_6^{(1)}$-type) third Painlev\'e equation in the space of initial values as the independent variable approaches infinity (or zero) and show that the limit set of each solution is compact and connected. Moreover, we prove that any solution with essential singularity at infinity has an infinite number of poles and zeroes, and similarly at the origin.