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Zbl 1161.33006
Wong, Roderick S.C.; Zhang, H.Y.
On the connection formulas of the third Painlevé transcendent. (English)
[J] Discrete Contin. Dyn. Syst. 23, No. 1-2, 541-560 (2009). ISSN 1078-0947; ISSN 1553-5231/e

The article reproduces the known connection formula for the parameters describing the asymptotic behavior of generic solutions of the SG-PIII equation, $u_{xx}+u_x/x+\sin u=0$, as $x\to0$ and $x\to+\infty$. Similarly to an earlier derivation by Novokshenov, the authors study the direct monodromy problem for a linear ODE associated with SG-PIII. However, they do not apply the conventional WKB analysis of the linear ODE which involves a matching procedure for a Liouville-Green approximation outside a neighborhood of a double turning point and a Weber-Hermit approximation in a vicinity of this double turning point. Instead, the authors use the so-called ``uniform asymptotics'' method by Clarkson, Bassom, Law and McLeod which extends the Weber-Hermite approximation globally using an appropriate change of variables.
[Andrei A. Kapaev (St. Petersburg)]

MSC 2000:: *33E17 Painlevé-type functions
34M55 Painlevé and other special equations
34M30 Asymptotics, summation methods
34M40 Stokes phenomena and connection problems

Keywords: Painlevé equation; connection formulas; direct monodromy problem

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