Quasi-linear Stokes phenomenon for the second Painlevé transcendent

Using the Riemann–Hilbert approach, we study the quasi-linear Stokes phenomenon for the second Painlevé equation y_xx = 2y³+xy−α. The precise description of the exponentially small jump in the dominant solution approaching α/x as |x|→∞ is given. For the asymptotic power expansion of the dominant solution, the coefficient asymptotics is computed.