On the Logarithmic Asymptotics of the Sixth Painleve' Equation (Summer 2007)

TL;DR

This paper analyzes the asymptotic behavior of solutions to the sixth Painleve' equation near critical points, using monodromy methods to characterize solutions' properties.

Contribution

It introduces a method to compute the monodromy group for solutions with logarithmic asymptotics and links asymptotic behavior directly to monodromy data.

Findings

Computed monodromy groups for solutions with logarithmic asymptotics

Characterized asymptotic behavior in terms of monodromy data

Provided a framework for understanding Painleve' solutions near critical points

Abstract

We study the solutions of the sixth Painlev\'e equation with a logarithmic asymptotic behavior at a critical point. We compute the monodromy group associated to the solutions by the method of monodromy preserving deformations and we characterize the asymptotic behavior in terms of the monodromy itself.