On the connection formulas of the third Painlevé transcendent (Q1001800)

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.