Connection formulas for the Ablowitz-Segur solutions of the inhomogeneous Painlev\'e II equation

TL;DR

This paper rigorously derives asymptotic behaviors and connection formulas for Ablowitz-Segur solutions of the inhomogeneous Painlevé II equation using Riemann-Hilbert problem techniques.

Contribution

It provides the first rigorous proof of asymptotics and connection formulas for these solutions, including pole-free regions for certain parameter ranges.

Findings

Asymptotics as x approaches ± infinity are established.

Connection formulas linking behaviors at different infinities are derived.

Real Ablowitz-Segur solutions have no real poles for α in (-1/2, 1/2).

Abstract

We consider the second Painlev\'e 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 $\alpha \in (-1/2, 1/2)$.