A Review of the Sixth Painlevé Equation

Abstract

For the Painlevé 6 transcendents, we provide a unitary description of the critical behaviours, the connection formulae, their complete tabulation, and the asymptotic distribution of poles close to a critical point.Note: This paper is published in Constr. Approx. 41 (2015), no. 3, 495527. In the present version, a few typing mistakes have been corrected, together with other small details.

Truncated Solutions of Painlevé Equation PV

“…Existence of truncated solution fo P III and P IV was re-established in [30] following methods in [22], and using a different method in [38], where the location of the first array of poles was also found. An overview of P VI is contained in [17]; see also [8]. The truncated solutions of the fifth Painlevé equation are the subject of the present article.…”

Resurgence, Painlevé equations and conformal blocks

“…Then the Heun equation takes the Schrödinger-like form: It is also possible to convert the PVI equation to a Weierstrass elliptic form by the following transformations [30,111,112]. Change the variable from t to τ , and the function from y(t)to u(τ ), as follows:…”

Isomonodromy, Painlevé transcendents and scattering off of black holes

“…These go by the name of isomonodromic deformations [10,[25][26][27]. For 4 regular singular points, these are known to reduce to the study of Painlevé transcendents, and many results about the latter came about from the study of this integrable structure [28][29][30]. In the following sections we outline the application of these techniques to solve the scattering of scalar fields around black holes.…”

“…This is the more general second order differential equation of the formz = R(z,ż, t), with R a rational function, which has the Painlevé property: the singularities of λ(t), apart from t = 0, 1, ∞, are simple poles and depend on the choice of initial conditions. Given a particular set of initial conditions, the equation can then be used to define a new transcendental function, the Painlevé transcendent P V I (θ ∞ , θ 0 , θ 1 , θ t ; t), in the same way the linear second order ordinary equation with 3 regular singular points can be used to define the hypergeometric function [10,30]. Now we see how the theory of isomonodromic deformations can help us to solve our initial scattering problem: Painlevé VI asymptotics are given in terms of the monodromy data of (3.11).…”