A tau-function solution to the sixth Painleve transcendent

TL;DR

This paper provides an explicit tau-function representation of the sixth Painleve transcendent, connecting it with elliptic functions and algebraic curve uniformization, offering new analytical tools for understanding its solutions.

Contribution

It introduces a tau-function solution expressed via elliptic integrals and theta functions, linking Painleve VI solutions to algebraic curve uniformization for the first time.

Findings

Explicit tau-function formulas in terms of elliptic functions.

Connection between Painleve VI and algebraic curve uniformization.

Examples illustrating the theoretical results.

Abstract

We represent and analyze the general solution of the sixth Painleve transcendent in the Picard-Hitchin-Okamoto class in the Painleve form as the logarithmic derivative of the ratio of certain $\tau$-functions. These functions are expressible explicitly in terms of the elliptic Legendre integrals and Jacobi $\theta$-functions, for which we write the general differentiation rules. We also establish a relation between the P6-equation and the uniformization of algebraic curves and present examples.