On the Critical Behavior, the Connection problem and the Elliptic representation of a Painleve VI equation (Spring 2001)

TL;DR

This paper investigates a broad class of solutions to the sixth Painleve' equation, analyzing their critical behavior, solving the connection problem, and exploring elliptic representations, with implications for WDVV equations.

Contribution

It introduces a new class of solutions covering nearly all monodromy data and provides detailed analysis of their critical behavior and connection problem.

Findings

Class of solutions covering almost all monodromy data

Explicit critical behavior near critical points

Solution of the connection problem

Abstract

We find a class of solutions of the sixth Painleve' equation appearing in the theory of WDVV equations. This class covers almost all the monodromy data associated to the equation, except one point in the space of the data. We find the critical behavior close to the critical points in terms of two parameters and solve the connection problem. We also study the critical behavior of Painleve' transcendents in the Elliptic representation.