Elliptic asymptotic representation of the fifth Painlev\'e transcendents

TL;DR

This paper presents an elliptic asymptotic representation of the fifth Painlevé transcendents near infinity, linking the main behavior to monodromy data and correcting previous results on Stokes graphs.

Contribution

It introduces a new elliptic asymptotic form for fifth Painlevé transcendents and clarifies the role of monodromy data in their phase shift dependence.

Findings

Asymptotic representation by Jacobi sn-function near infinity

Main part depends on monodromy data as a phase shift

Corrections provided for Stokes graph and earlier results

Abstract

For the fifth Painlev\'e transcendents an asymptotic representation by the Jacobi $\mathrm{sn}$-function is presented in cheese-like strips along generic directions near the point at infinity. Its elliptic main part may be understood to depend on the phase shift as a single integration constant, which is parametrised by monodromy data for the associated isomonodromy deformation. The other integration constant is contained in the error term or in a correction function. This paper contains corrections of the Stokes graph and of the related results in the early version.