Asymptotic properties of a family of solutionsof the Painlevé equation PVI

Abstract

In analyzing the question whether nonlinear equations can define new functions with good global properties, Fuchs had the idea that a crucial feature now known as the Painleve property (PP) is the absence of movable (meaning their position is solutiondependent) essential singularities, primarily branch-points, see [8]. First order equations were classified with respect to the PP by Fuchs, Briot and Bouquet, and Painleve by 1888, and it was concluded that they give rise to no new functions. Painleve and Gambier took this analysis to second order, looking for all equations of the form u ′′ = F(u ′, u, z), with F rational in u ′, algebraic in u, and analytic in z, having the PP [18, 19]. They found some fifty types with this property and succeeded to solve all but six of them in terms of previously known functions. The remaining six types are now known as the Painleve equations. Beginning in the 1980s, almost a century after their discovery, these equations were related to linear problems (and thereby solved) by various methods including the powerful techniques of isomonodromic deformation and reduction to Riemann-Hilbert problems [3, 4, 7, 11]. The solutions of the six Painleve equations play a fundamental role in many areas of pure and applied mathematics due to their integrability properties. In particular, there are numerous physical applications of the Painleve PVI equation (for some references see, e.g., [6]) among which we mention the problem of construction of self-dual Bianchi-type IX Einstein metrics, [2, 5, 17, 21]