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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 =2y 3 +xy-α

by using the Riemann-Hilbert approach. The main result is the precise description of the exponentially small jump of the dominant solution approaching α/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:
34M40Stokes phenomena and connection problems (ODE in the complex domain)
34M50Inverse problems in theory of ODE in the complex domain (Riemann-Hilber, etc.)
34M55Painlevé and other special equations; classification, hierarchies