Three-Parameter Solutions of the PV Schlesinger-Type Equation near the Point at Infinity and the Monodromy Data

TL;DR

This paper constructs a three-parameter family of solutions to a Schlesinger-type equation related to Painlevé V near infinity, providing explicit monodromy data and analyzing zero and pole sequences.

Contribution

It introduces a new three-parameter family of solutions and their monodromy data for the Schlesinger-type equation linked to Painlevé V, expanding understanding of its solution space.

Findings

Explicit three-parameter solutions near infinity

Monodromy data derived via matching techniques

Solutions exhibit zeros and poles along the imaginary axis

Abstract

For the Schlesinger-type equation related to the fifth Painlev\'e equation (V) via isomonodromy deformation, we present a three-parameter family of matrix solutions along the imaginary axis near the point at infinity, and also the corresponding monodromy data. Two-parameter solutions of (V) with their monodromy data immediately follow from our results. Under certain conditions, these solutions of (V) admit sequences of zeros and of poles along the imaginary axis. The monodromy data are obtained by matching techniques for a perturbed linear system.