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 $\Psi$-function corresponding to the solution $y\sim\pm\sqrt{-x/2}$ as $|x|\to\infty$ 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.