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On the connection formulas of the third Painlevé transcendent. (English) Zbl 1161.33006
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.
33E17Painlevé-type functions
34M55Painlevé and other special equations; classification, hierarchies
34M30Asymptotics, summation methods (ODE in the complex domain)
34M40Stokes phenomena and connection problems (ODE in the complex domain)
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