Asymptotic Properties of a Family of Solutions of the Painlevé Equation P VI

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 Painlevé property (PP) is the absence of movable (meaning their position is solution-dependent) essential singularities, primarily branch-points, see [8]. First order equations were classified with respect to the PP by Fuchs, Briot and Bouquet, and Painlevé by 1888, and it was concluded that they give rise to no new functions. Painlevé 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 [17, 18]. 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 Painlevé equations. Beginning in the 1980’s, 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], [15]. The solutions of the six Painlevé 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 Painlevé 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, 16, 19] the classification of the solutions of WDVV equation in 2D-topological field theories and probability, theory, especially random matrix theory (see e.g. [20], [21]). A three parameter family of solutions of the Painlevé equation PVI arises in the context of random matrix theory in a recent work of Borodin and Deift [1]. The asymptotic behavior of the solutions of the Painlevé equations is of utmost importance. The main purpose of this paper is to characterize a family of solutions of PVI for large argument, relevant to the study [1] In the σ-form these solutions satisfy (see [9]—eq. (C.61), with ν1 = ν2):