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

We consider the second Painlevé equation u'' (x) = 2u³ (x) + xu (x) − α, where α is a nonzero constant. Using the Deift–Zhou nonlinear steepest descent method for Riemann–Hilbert problems, we rigorously prove the asymptotics as x → ±∞ 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 α ∈ (-1/2, 1/2) .