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

TL;DR

This paper investigates the long-term behavior of solutions to the fourth Painlevé equation in complex initial value space, revealing properties of their limit sets and the distribution of poles and zeros.

Contribution

It provides a detailed analysis of the asymptotic behavior of Painlevé IV solutions in the extended initial value space, extending previous phase space descriptions.

Findings

Limit sets of solutions are compact and connected.

Non-special solutions have infinitely many poles.

Non-special solutions also have infinitely many zeros.

Abstract

We study the asymptotic behaviour of solutions of the fourth Pain\-lev\'e equation as the independent variable goes to infinity in its space of (complex) initial values, which is a generalisation of phase space described by Okamoto. We show that the limit set of each solution is compact and connected and, moreover, that any non-special solution has an infinite number of poles and infinite number of zeroes.