Quasi-linear Stokes phenomenon for the Hastings-McLeod solution of the second Painlev\'e equation

Using the Riemann-Hilbert approach, we explicitly construct the asymptotic $\mathrm{\Psi }$-function corresponding to the solution $y\sim \pm \sqrt{-x\mathrm{/}2}$ as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$ to the second Painlev'e 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.