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Quasi-linear Stokes phenomenon for the second Painlevé transcendent. (English) Zbl 1048.34146

The paper is devoted to the study of the quasi-linear Stokes phenomenon for the second Painlevé equation

${y}_{xx}=2{y}^{3}+xy-\alpha$

by using the Riemann-Hilbert approach. The main result is the precise description of the exponentially small jump of the dominant solution approaching $\alpha /x$ as $|x|$ tends to infinity. The paper closes with the computation of the coefficient asymptotics for the asymptotic power expansion of the dominant solution.

##### MSC:
 34M40 Stokes phenomena and connection problems (ODE in the complex domain) 34M50 Inverse problems in theory of ODE in the complex domain (Riemann-Hilber, etc.) 34M55 Painlevé and other special equations; classification, hierarchies