Asymptotic expansions of solutions of the sixth Painlevé equation

Abstract

We obtain all asymptotic expansions of solutions of the sixth Painlevé equation near all three singular points x = 0, x = 1, and x = ∞ for all values of four complex parameters of this equation. The expansions are obtained for solutions of five types: power, power-logarithmic, complicated, semiexotic, and exotic. They form 117 families. These expansions may contain complex powers of the independent variable x. First we use methods of two-dimensional power algebraic geometry to obtain those asymptotic expansions of all five types near the singular point x = 0 for which the order of the leading term is less than 1. These expansions are called basic expansions. They form 21 families. All other asymptotic equations near three singular points are obtained from basic ones using symmetries of the equation. The majority of these expansions are new. Also, we present examples and compare our results with previously known ones. Introduction In 1884–1885 L. Fuchs [47] and H. Poincaré [70, 71, 72] suggested looking for differential equations whose solutions do not have movable critical points and cannot be expressed in terms of previously known functions. In 1889 S. Kovalevskaya [36] had shown that the absence of movable critical points of solutions allows one to construct solutions analytically. A singular point x = x0 of a function y(x) of a complex variable x is called a critical singular point if the value of the function y(x) changes as x moves along the path surrounding x0. A movable singular point of a solution of a differential equation is a singular point such that its position depends on initial conditions of the problem. For example for the solution y = 1/ √ x− x0, where x0 is an arbitrary constant, the point x = x0 is a movable critical point. By a meromorphic function we mean a function whose only singularities in the finite part of the complex place are poles. In 1887 E. Picard [68] suggested studying the following class of second order equations: (1) y′′ = F (x, y, y′), where the function F is rational in y and y′ and meromorphic in x, and to find, among equations (1), those that have only immovable critical singular points. At the beginning of the 20th century P. Painlevé [65, 66, 67], and his students B. Gambier [49] and R. Garnier [50, 51] solved the problem formulated by Fuchs and Picard. They have found 50 canonical equations of the form (1) whose solutions have no movable critical singular 2010 Mathematics Subject Classification. Primary 34E05; Secondary 34M55. The work was supported by the Russian Foundation of Fundamental Research (Project 08–01–00082) and the Foundation for the Assistance to Russian Science. Editorial Note: The following text incorporates changes and corrections submitted by the authors for the English translation. c ©2010 American Mathematical Society 1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2 A. D. BRUNO AND I. V. GORYUCHKINA points. Solutions of 44 of these equations could be expressed in terms of known (elementary or special) functions, and solutions of the remaining six equations determine new special functions, which are now called Painlevé transcendents. The sixth Painlevé equation first appeared in the paper by R. Fuchs [48]. In has the form y′′ = (y′)2 2 ( 1 y + 1 y − 1 + 1 y − x ) − y′ ( 1 x + 1 x− 1 + 1 y − x ) + y(y − 1)(y − x) x2(x− 1)2 [ a+ b x y2 + c x− 1 (y − 1)2 + d x(x− 1) (y − x)2 ] , (2) where a, b, c, d are complex parameters, x and y are complex variables, y′ = dy dx . It has three singular points x = 0, x = 1 and x = ∞, and is usually denoted P6. E. Picard [69] has found solutions of this equation in an explicit form for special values of four parameters: a = b = c = 0, d = 12 . R. Garnier [51] was studying solutions of this equation without any restrictions on parameters. A new wave of interest to Painlevé equations occurred in the 1970s after M. Ablowitz, A. Ramani, and H. Segur [1, 42, 43] discovered that integrable nonlinear partial differential equations are related to Painlevé equations (see also [35, 37]). For example, the sixth Painlevé equation is a reduction of the Ernst equation in general relativity. Nowadays, the followng problems for the Painlevé equations are being studied: asymptotic behavior of solutions near singular points, local and global properties of solutions, rational and algebraic solutions, discretization, applications of Painlevé equations (mainly in physics). In the present paper we study asymptotic expansions of solutions of the sixth Painlevé equation at the singular points x = 0, 1, ∞. Expansions in nonsingular points were described in [54, § 46], and, using power geometry, in [13, 24]. Similar studies were performed by many authors. S. Shimomura [73]–[76], M. Jimbo [61], H. Kimura [62], K. Okamoto [64] proved, using a variety of methods, existence and convergence of twoparameter families of expansions for solutions of the sixth Painlevé equation. In the book [54, § 46] by I. V. Gromak, I. Laine, and S. Shimomura the authors describe asymptotic expansions of solutions in integer powers of the independent variable. For some special values of parameters of the sixth Painlevé equation, B. Dubrovin and M. Mazzocco [46, 63], and also D. Guzzetti [55] obtained several initial terms of nonpower and exotic expansions. A comparison of these results with ours is presented at the end of the Introduction and in Section 2 of Chapter 4. The study of asymptotic extensions and asymptotic properties of solutions of Painlevé equations near singular points consists of the following three main steps. Step 1. To find formal solutions in the form of asymptotic expansions