Meromorphic Painlevé transcendents at a fixed singularity

Abstract

We classify solutions of the third Painlevé equation P III , which are meromorphic around the origin, and determine their linear monodromy. Combined with [4], [5], all of meromorphic solutions of the Painlevé equations around fixed singular points of the Briot-Bouquet type are completely determined. We characterize the linear monodromy of meromorphic solutions for the third, fifth and sixth Painlevé equations.

Geometric Aspects of the Painlevé Equations PIII(D6) and PIII(D7)

“…Observation. After scaling some variables one sees that M(θ) is the union of two open affine spaces U 1 × T and U 2 × T , where U 1 is given by the variables a 1 , b 1 , c 0 and the relation a 2 1 + (1 − a 1 − b 1 c 0 )c 0 = 0, and U 2 is given by the variables a −1 , b 0 , c −1 and the relation…”

“…3. For α = ±1, the reducible locus S(α, α) * red of S(α, α) * is represented by the union of the two families in (2) given by e 2πid = α, 1 = 2 = 1 and 1 = 2 = −1. Each of the two families is isomorphic to P 1 × T , by sending the matrix differential operator to ((c 1 : c 0 ), t) ∈ P 1 × T .…”

“…Isomonodromy for reducible connections produces Riccati solutions and resonance is related to algebraic solutions. 2 The family (1, −, 1/2) and PIII(D 7 )…”

“…This leads to a special solution q for PIII(D 7 ) and θ ∈ 1 2 + Z, which is transcendental according to [5]. According to [2], q is a univalent function of t and is a meromorphic at t = 0.…”

Symmetric solutions to the four dimensional degenerate Painlevé type equation NY A 4

“…Based on A.V. Kitaev's idea, we have studied special solutions with generic values of parameters for the fourth, fifth, sixth and third Painlevé equations, for which the linear monodromy can be calculated explicitly [10][11][12][13].…”