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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. (English) Zbl 1064.34078

The authors find the asymptotics and the connection formulae as $\tau \to ±0,±i0,±\infty ,±i\infty$ 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

${y}^{\text{'}\text{'}}={\left({y}^{\text{'}}\right)}^{2}/y-{y}^{\text{'}}/\tau +\left(\alpha {y}^{2}+\beta \right)/\tau +\gamma {y}^{3}+\delta /y·$

The result is obtained by implementing the isomonodromy deformation technique developed in the midle of 1980s by [A. R. Its and 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.

##### MSC:
 34M55 Painlevé and other special equations; classification, hierarchies 34M40 Stokes phenomena and connection problems (ODE in the complex domain)