Connection formulas for the Ablowitz–Segur solutions of the inhomogeneous Painlevé II equation

Abstract

We consider the second Painlevé equationu′′(x)=2u3(x)+xu(x)−α,where is a nonzero constant. Using the Deift–Zhou nonlinear steepest descent method for Riemann–Hilbert problems, we rigorously prove the asymptotics as for both the real and purely imaginary Ablowitz–Segur solutions, as well as the corresponding connection formulas. We also show that the real Ablowitz–Segur solutions have no real poles when .