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 $\Psi$-function corresponding to the solution $y\sim \pm \sqrt{-x/2}$ as $|x|\to \infty$ to the second Painlevé equation $y_{xx}=2y^3+xy-\alpha$. We precisely describe the exponentially small jump in the dominant solution and the coefficient asymptotics in its power-like expansion.

LaTeX2e, 22 pages