Asymptotic and monodromy problems for higher-order Painlev\'e III equations

TL;DR

This paper investigates the asymptotic behavior and monodromy data of higher-order Painlevé III equations through isomonodromy deformations, providing explicit formulas and applications to tt* equations.

Contribution

It introduces a detailed analysis of the asymptotics and monodromy for higher-order Painlevé III equations, including explicit formulas for Stokes and connection matrices.

Findings

Explicit formulas for Stokes matrices and connection matrices.

Parameterization of solutions via asymptotic parameters.

Application to the study of tt* equations.

Abstract

In this paper, we study the isomonodromy deformation equations for the $n\times n$ system of first order meromorphic linear ordinary differential equations with two second order poles. We analyze the asymptotic behaviour of the solutions at a boundary point of the isomonodromic deformation space, and derive a parameterization of the solutions via asymptotic parameters. We then derive the explicit formula for the Stokes matrices and connection matrix of the associated linear system in terms of the asymptotic parameters. In the end, we apply the results to the study of the $tt^{*}$ equations.