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Zbl 1064.34078
Kitaev, A.V.; Vartanian, A.H.
Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. (English)
[J] Inverse Probl. 20, No. 4, 1165-1206 (2004). ISSN 0266-5611

The authors find the asymptotics and the connection formulae as $\tau\to\pm0,\pm i0,\pm\infty,\pm i\infty$ for a class of regular at infinity solutions of the degenerate third Painlevé equation. The latter can be thought of as the special case ($\gamma=0$) of the general PIII equation $$y''=(y')^2/y-y'/\tau+\bigl(\alpha y^2+\beta\bigr)/\tau+\gamma y^3+\delta/y.$$ The result is obtained by implementing the isomonodromy deformation technique developed in the midle of 1980s by [{\it A.~R. Its} and {\it V.~Yu. Novokshenov}, The isomonodromic deformation method in the theory of Painlevé equations. Lecture Notes in Mathematics. 1191. Berlin: Springer-Verlag, (1986; Zbl 0592.34001)]. Namely, the authors introduce a suitable Lax pair for this equation and describe its manifold of the monodromy data. Then, imposing certain assumptions on the coefficients of the Lax pair, they apply an elementary WKB analysis in order to solve the direct monodromy problem, asymptotically as $\tau\to+\infty$ and $\tau\to+0$, i.e., to find the asymptotic formulae for the relevant monodromy data in terms of the coefficients of the Lax pair. Finally, inverting the obtained expressions, the authors find the asymptotics of the Painlevé function in terms of the monodromy data. The results for other directions in the complex $\tau$-plane follow from the symmetries of the equation and the Lax pair.
[Andrei A. Kapaev (St. Petersburg)]

MSC 2000:: *34M55 Painlevé and other special equations
34M40 Stokes phenomena and connection problems

Keywords: Painlevé equation; isomonodromy deformations; connection formulae

Citations: Zbl 0592.34001

Cited in: Zbl 1210.34130

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