Elliptic asymptotics for the complete third Painlev\'e transcendents

TL;DR

This paper investigates the asymptotic behavior of solutions to the third Painlevé equation using the Boutroux ansatz, expressing solutions in terms of Jacobi functions along specific complex directions.

Contribution

It provides a novel asymptotic representation of Painlevé III solutions via isomonodromy deformation and WKB analysis, extending understanding of their complex structure.

Findings

Asymptotic solutions expressed with Jacobi sn-function

Analysis of Stokes curves on Riemann surfaces

Application of Boutroux ansatz to Painlevé III

Abstract

For a general solution of the third Painlev\'e equation of complete type we show the Boutroux ansatz near the point at infinity. It admits an asymptotic representation in terms of the Jacobi sn-function in cheese-like strips along generic directions. The expression is derived by using isomonodromy deformation of a linear system governed by the third Painlev\'e equation of this type. In our calculation of the WKB analysis, the treated Stokes curve ranges on both upper and lower sheets of the two sheeted Riemann surface.