New transformations for Painlevé’s third transcendent

Abstract

Abstract. We present transformations relating the third transcendent of Painlevé with parameter sets located at the corners of the Weyl chamber for the symmetry group of the system, the affine Weyl group of the root system B (1) 2 , to those at the origin. This transformation entails a scaling of the independent variable, and implies additive identities for the canonical Hamiltonians and product identities for the τ -functions with these parameter sets.A curious anomaly has existed in the theory of the Painlevé transcendents for many years, for almost a century in fact. This anomaly is the existence of a particular transformation for the second Painlevé transcendent q(t; α + 1 / 2 ) that had no known analogue for any of the other transcendents. One recalls that the second Painlevé transcendent q = q(t; α + 1 / 2 ) is by definition a solution of the P II differential equation 1 , and usingtransformations of this group, solutions for arbitrary v 1 can be related to a solution whose parameter v 1 ∈ (0, 1 / 2 ] (or in [ 1 / 2 , 1)). This interval defines the Weyl chamber and the above transformation in question relates the solutions at the endpoints of this interval (see Figure 1).Specifically the transformation takes the form

Bäcklund flux quantization in a model of electrodiffusion based on Painlevé II

“…We note that (41) can also be obtained from (39) using (23). In each of these three solutions, the concentrations are everywhere positive as required.…”

Parametrization of solutions of the Lewis metric by a Painlevé transcendent III

“…6 ͓see Eq. 12 Setting Furthermore, ͑22͒ corresponds to the standard form of the V transcendent of Painlevé as proposed by Okamoto 11 with the parameter ␤ = 0, i.e., v 1 + v 2 = 0 and t = x.…”

Isomonodromy for the Degenerate Fifth Painlevé Equation

“…Classical' papers on the subject are [8,9,12,13,14,15,16]. Especially relevant for the present text are the paper by Ohyama and Okumura [11], Witte's paper [26]. The book [2] by Fokas, Its, Kapaev, and Novokshenov discusses more analytic aspects of the Riemann-Hilbert correspondence, but it does not discuss the degenerate fifth Painlevé equation.…”

“…Now we compare the group of the Bäcklund transformations for degP V with the work of N.S. Witte [26]. We restrict our transformations to the case of even solutions of degP V and find the classical P V (22), (23) of [26] lead to v 1 = θ 0 + θ 1 and v 2 = θ 0 − θ 1 .…”

“…Witte [26]. We restrict our transformations to the case of even solutions of degP V and find the classical P V (22), (23) of [26] lead to v 1 = θ 0 + θ 1 and v 2 = θ 0 − θ 1 . Our group of transformations for the even solutions of degP V is generated by (θ 0 , θ 1 ) → (θ 0 + 1, θ 1 ) and (θ 0 , θ 1 ) → (θ 1 , θ 0 ) and the 'trivial' transformations (θ 0 , θ 1 ) → (±θ 0 , ±θ 1 ).…”