Quasi-linear Stokes phenomenon for the second Painlevé transcendent

Abstract

Using the Riemann-Hilbert approach, we study the quasi-linear Stokes phenomenon for the second Painlevé equation $y_{xx}=2y^3+xy-α$. The precise description of the exponentially small jump in the dominant solution approaching $α/x$ as $|x|\to\infty$ is given. For the asymptotic power expansion of the dominant solution, the coefficient asymptotics is found.