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

Abstract

We consider the second Painlevé equation $$u"(x)=2u^3(x)+xu(x)-\alpha ,$$ where $\alpha$ is a nonzero constant. Using the Deift-Zhou nonlinear steepest descent method for Riemann-Hilbert problems, we rigorously prove the asymptotics as $x\to \pm \infty$ 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 $\alpha \in (-1/2,1/2)$.

minor revision, 32 pages, 8 figures

Cited by in corpus (2)

• Singular asymptotics for solutions of the inhomogeneous Painlevé II equation
• Mesoscopic critical fluctuations