The nonlinear steepest descent approach to the singular asymptotics of the second Painlevé transcendent

We consider the real-valued solutions of the second Painlevé equation on the real line. The two-parameter family of the solutions having singular asymptotics as x → + ∞ or/and x → − ∞ is studied with the help of the Deift–Zhou nonlinear steepest descent method. Explicit evaluation in terms of trigonometric functions of the (singular) leading orders of the asymptotics is carried out and the corresponding connection formulae obtained. A novel methodological feature is the appearance of the “soliton” type Riemann–Hilbert problem in the course of the implementation of the Deift–Zhou scheme for the Riemann–Hilbert problem corresponding to the second Painlevé equation within the Riemann–Hilbert isomonodromy approach. The result of the paper reproduces previously known formulae derived by Kapaev via the original isomonodromy technique.