Application of uniform asymptotics to the connection formulas of the fifth Painlevé equation

Abstract

We apply the uniform asymptotics method proposed by Bassom, Clarkson, Law and McLeod [4] to a special Painlevé V equation, and we provide a simpler and more rigorous proof of the connection formulas for a special solution of the equation, which have been established earlier by McCoy and Tang via the isomonodromy and WKB methods.

A note on the connection problem of some special Painlevé V functions

“…Recently, the method of "uniform asymptotics"has been applied to the connection problems for PV, cf. Zeng and Zhao [13]. However, all these cases also differ from our present case.…”

“…Although this part is slightly different from the approach in [3], they are coincide essentially. See details in Sec.6 of [3] or the corresponding sections in [11][12][13]. Here we only give the derivation of s 1 .…”

Irregular conformal blocks and connection formulae for Painlevé V functions

“…The connection problems between 0, +∞ and i ∞ were also studied in[MT1,MT2,MT3] and more recently in e.g [AK2,DKV,ZZ,. LZZ] for special classes of PV solutions with constrained parameters θ 0,t , * and non-generic initial conditions.…”

On the connection problem for nonlinear differential equation

“…While the difficulty in extending the techniques for PII to other transcendents is also acknowledged by the authors of [1, p.244]. Yet the method has been applied to the connection problems by Wong and Zhang [17,18], Zeng and Zhao [19], Long, Zeng and Zhou [10]. Recently, Long el at.…”