The logarithmic asymptotics of the sixth Painlevé equation

Abstract

We compute the monodromy group associated to the solutions of the sixth Painlevé equation with a logarithmic asymptotic behavior at a critical point and we characterize the asymptotic behavior in terms of the monodromy itself.

Chapter 18. Expansions for Solutions of the Schlesinger Equation at a Singular Point

“…Corollary 1 and (3) give the power expansions for solutions of the sixth Painlevé equation. A different asymptotics for sixth Painlevé equation was obtained in D. Guzzetti [2], A. Bruno and I. Goryuchkina [5], M. Mazzocco [3] and others. For the sixth Painlevé equation, we have an analogue of Corollary 1.…”

“…Notice that the measure of the systems of non-general case is equal to zero. Now consider the case n = 4, p = 2, which is equivalent to case of the sixth Painlevé equation (2). Without loss of generality, let us fix three variable a 1 = 0, a 2 = 1, a 3 = ∞ and denote a 4 by t. We obtain the system of ordinary differential equations with variable t and unknown matrix-functions…”

A Review of the Sixth Painlevé Equation

“…f is generically injective, according to the following Proposition 1 Let the order of loops be fixed. The map (14) is injective (one-to-one) when [20].…”

“…The effect of the bi-rational transformation on the monodromy data p ij is described in [29], [10], [20], and more generally in [11] (see also [38]). In particular, we have:…”

Connection formulas for the $\lambda$ generalized Ising correlation functions

“…However, the Painlevé VI for the Ising model is not generic and therefore the results of [9], while still relevant at t = 1 where the correlation function is singular, do not hold at t = 0, ∞ where the correlation function is analytic. Nongeneric cases have been studied by Guzzetti [11]) but the case relevant for the generalized Ising correlations seems not to have been investigated.…”