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

TL;DR

This paper investigates the global properties of tronquée and tritronquée solutions to the third and fourth Painlevé equations, including their sectors of analyticity, Borel summed representations, and singularity positions.

Contribution

It provides comprehensive analysis of these special solutions, including sectorial analyticity, Borel summability, and precise asymptotic singularity locations, extending previous results.

Findings

Identified sectors of analyticity for the solutions.

Derived Borel summed representations in these sectors.

Located the asymptotic positions of singularities near sector boundaries.

Abstract

Recently in a paper by Lin, Dai and Tibboel, it was shown that the third and fourth Painlev\'e equations have tronqu\'ee and tritronqu\'ee solutions. We obtain global information about these tronqu\'ee and tritronqu\'ee solutions. We find their sectors of analyticity, their Borel summed representations in these sectors as well as the asymptotic position of the singularities near the boundaries of the analyticity sectors. We also correct slight errors in the paper mentioned.