Solving the Sixth Painlevé Equation: Towards the Classification of all the Critical Behaviors and the Connection Formulae

Abstract

Abstract:The critical behavior of a three real parameter class of solutions of the sixth Painlevé equation is computed, and parametrized in terms of monodromy data of the associated 2 × 2 matrix linear Fuchsian system of ODE. The class may contain solutions with poles accumulating at the critical point. The study of this class closes a gap in the description of the transcendents in one to one correspondence with the monodromy data. These transcendents are reviewed in the paper. Some formulas that relate the monodromy data to the critical behaviors of the four real (two complex) parameter class of solutions are missing in the literature, so they are computed here. A computational procedure to write the full expansion of the four and three real parameter class of solutions is proposed.

Tabulation of Painlevé 6 transcendents

“…Let (θ 0 , θ 1 , θ x , θ ∞ ) ∈ C 4 be fixed by PVI, up to the equivalence θ k → −θ k , k = 0, x, 1, and θ ∞ → 2 − θ ∞ . Denote ∼ the equivalence and let For an equivalent definition, see section 2 of [16].…”

“…Jimbo's work [21] is the foundation of all subsequent developments. In this paper we use the results of [21], [3], [12], [13], [14], [15], [16] to provide behaviours (7) in Tables 1, 2 and 3, and the formulae of type (8) and (9) in Section 5 . Tables 1,2, and 3 of Critical behaviors Are Tables 1, 2 and 3 complete, namely do they provide all the critical behaviours?…”

“…In other cases, the formulae can be computed from the cases ⋆ via a birational transformation. [16]). In the case of (36), the c nm 's depend rationally also on σ.…”

“…When p 0x = 2 cos π(θ ∞ + θ 1 ), the Jimbo-Fricke cubic has factorisation (16) and σ = ±(θ ∞ + θ 1 ) + 2n ± for suitable integers n ± such that −1 < ℜσ < 1 (or ℜσ > −1 in case of (41)). The same considerations as above apply to the limit of a in (29) for σ → ±(θ ∞ + θ 1 ) + 2n.…”

Analytic solutions ofq-P(A1) near its critical points

“…Guzzetti [10] states that the full expansion of the solution ω(t) of P VI , characterised by equation (7.22), is given by…”