A Review on The Sixth Painleve' Equation

TL;DR

This paper provides a comprehensive review of the sixth Painleve' equation, detailing its critical behaviors, connection formulas, pole distribution, and tabulation for better understanding of its solutions.

Contribution

It offers a complete and unified overview of the critical behaviors, connection formulas, and pole distribution of Painleve' 6 transcendents, consolidating existing knowledge.

Findings

Detailed description of critical behaviors near singularities

Complete tabulation of solutions and connection formulas

Asymptotic distribution of poles near critical points

Abstract

For the Painlev\'e 6 transcendents, we provide a unitary description of the critical behaviours, the connection formulae, their complete tabulation, and the asymptotic distribution of the poles close to a critical point.