On the solvability of the Painleve VI equation

Abstract

A rigorous method was introduced by Fokas and Zhou (1992) for studying the Riemann-Hilbert problem associated with the Painleve II and IV equations. The same methodology has been applied to the Painleve I, III and V equations. In this paper, we will apply the same methodology to the Painleve VI equation. We will show that the Cauchy problem for the Painleve VI equation admits, in general, a global meromorphic solution in t. Furthermore, the special solution which can be written in terms of a hypergeometric function is obtained via solving the special case of the Riemann-Hilbert problem.