Existence and Uniqueness of Tronqu\'ee Solutions of the Third and Fourth Painlev\'e Equations

TL;DR

This paper demonstrates the existence of pole-free tronquée solutions for the third and fourth Painlevé equations, extending known results from the first two equations and enriching the understanding of their asymptotic behaviors.

Contribution

It establishes the existence of tronquée solutions for the third and fourth Painlevé equations, a novel extension of prior work on the first two equations.

Findings

Existence of pole-free tronquée solutions for P III and P IV.

Extension of asymptotic solution theory to higher Painlevé equations.

Provides a foundation for further analysis of Painlevé solutions in complex sectors.

Abstract

It is well-known that the first and second Painlev\'e equations admit solutions characterised by divergent asymptotic expansions near infinity in specified sectors of the complex plane. Such solutions are pole-free in these sectors and called tronqu\'ee solutions by Boutroux. In this paper, we show that similar solutions exist for the third and fourth Painlev\'e equations as well.