Connection formulae for asymptotics of the fifth Painlevé transcendent on the real axis

Abstract

Leading terms of asymptotic expansions for the general complex solutions of the fifth Painlevé equation as t → ı∞ are found. These asymptotics are parameterized by monodromy data of the associated linear ODE,The parametrization allows one to derive connection formulas for the asymptotics. We provide numerical verification of the results. Important special cases of the connection formulas are also considered.

Critical Behaviours of the Fifth Painlevé Transcendents and the Monodromy Data

“…the proof of Theorem 2.10). In the expressions above it seems that the cancellations of the factors 2 occur, but we have not succeeded in proving them.…”

“…Applying WKB analysis to (1.1), Andreev and Kitaev [2] obtained asymptotic solutions of (V) along the real axis near x = 0 and x = ∞ together with their monodromy data that yield the connection formulas between them. For more general integration constants, a family of solutions near x = 0 expanded into convergent series in spiral domains or sectors was given by the present author [22].…”

“…Toward this goal, in this paper, near the critical point x = 0 of (V), we present families of solutions expanded into convergent series of three types and the respective monodromy data parametrized by integration constants yielding analogues of the parametric connection formulas in [11]. The monodromy data are given under more general conditions on θ 0 , θ x , θ ∞ and the integration constants than those of [2] or [14]. These solutions, including degenerate cases, are of complex power type, of inverse logarithmic type and of Taylor series type (cf.…”

“…Theorems 2.1 and 2.3-2.5). For these solutions, under certain conditions, it is possible to derive connection formulas relating to asymptotic solutions as x → +∞ in [2], and the total set of them is almost complete (cf. Sections 2.5 and 2.6).…”

Fifth Painleve Transcendents Which are Analytic at the Origin

“…For the fifth Painlevé equation, such a correspondence was given by Andreev and Kitaev [1] [2] by means of WKB analysis. Although the connection formula given by Andreev and Kitaev is complicated in general, we show in this paper that the monodromy data for such special solutions that are analytic at the origin can be determined explicitly by an elementary method.…”