Solving the Sixth Painleve' Equation: Towards the Classification of all the Critical Behaviours and the Connection Formulae (October 2010)

TL;DR

This paper analyzes a class of solutions to the sixth Painleve' equation, computing their critical behaviors and monodromy data, and provides formulas and procedures to describe their full expansions and connection formulae.

Contribution

It introduces a comprehensive method to classify and compute the critical behaviors and connection formulae of solutions to the sixth Painleve' equation based on monodromy data.

Findings

Computed critical behaviors for a three-parameter solution class.

Derived formulas relating monodromy data to solution behaviors.

Proposed a computational procedure for full solution expansions.

Abstract

The critical behavior of a three real parameter class of solutions of the sixth Painlev\'e equation is computed, and parametrized in terms of monodromy data of the associated $2\times 2$ matrix linear Fuchsian system of ODE. The class may contain solutions with poles accumulating at the critical point. The study of this class closes a gap in the description of the transcendents in one to one correspondence with the monodromy data. These transcendents are reviewed in the paper. Some formulas that relate the monodromy data to the critical behaviors of the four real (two complex) parameter class of solutions are missing in the literature, so they are computed here. A computational procedure to write the full expansion of the four and three real parameter class of solutions is proposed.