Trans-Series Asymptotics of Solutions to the Degenerate Painlev\'{e} III Equation: A Case Study

TL;DR

This paper develops a detailed trans-series asymptotic analysis of solutions to the Degenerate Painlevé III Equation using isomonodromy methods, revealing the structure of solutions and their symmetries.

Contribution

It introduces a parametrization of trans-series solutions via monodromy data and derives symmetry actions for the DP3E.

Findings

Parametrization of solutions in terms of monodromy data

Explicit trans-series asymptotics for Hamiltonian and auxiliary functions

Identification of Lie-point symmetries of the DP3E

Abstract

A one-parameter family of trans-series asymptotics of solutions to the Degenerate Painlev\'{e} III Equation (DP3E) are parametrised in terms of the monodromy data of an associated two-by-two linear auxiliary problem via the isomonodromy deformation approach: trans-series asymptotics for the associated Hamiltonian and principal auxiliary functions and the solution of one of the sigma-forms of the DP3E are also obtained. The actions of Lie-point symmetries for the DP3E are derived.