Matching Procedure for the Sixth Painlev\'e Equation (May 2006)

TL;DR

This paper introduces a constructive method to analyze the critical behavior of Painleve VI transcendents, solving the connection problem and classifying solutions with various behaviors.

Contribution

It provides a new constructive procedure to determine the critical behavior and connection problem solutions for Painleve VI transcendents, including families with different asymptotic behaviors.

Findings

Derived two- and one-parameter families of solutions

Classified solutions with trigonometric, logarithmic, and Taylor behaviors

Solved the connection problem for Painleve VI

Abstract

We present a constructive procedure to obtain the critical behavior of Painleve' VI transcendents and solve the connection problem. This procedure yields two and one parameter families of solutions, including trigonometric and logarithmic behaviors, and three classes of solutions with Taylor expansion at a critical point.