Quasi-linear Stokes phenomenon for the Hastings-McLeod solution of the second Painlevé equation

Abstract

Using the Riemann-Hilbert approach, we explicitly construct the asymptotic $Ψ$-function corresponding to the solution $y\sim\pm\sqrt{-x/2}$ as $|x|\to\infty$ to the second Painlevé equation $y_{xx}=2y^3+xy-α$. We precisely describe the exponentially small jump in the dominant solution and the coefficient asymptotics in its power-like expansion.