On the quasi-Ablowitz-Segur and quasi-Hastings-McLeod solutions of the inhomogeneous Painlevé II equation

Abstract

We consider the quasi-Ablowitz-Segur and quasi-Hastings-McLeod solutions of the inhomogeneous Painlevé II equation $$ u"(x)=2u^3(x)+xu(x)-α\qquad \textrm{for } α\in \mathbb{R} \textrm{ and } |α| > \frac{1}{2}. $$ These solutions are obtained from the classical Ablowitz-Segur and Hastings-McLeod solutions via the Bäcklund transformation, and satisfy the same asymptotic behaviors when $x \to \pm \infty$. For $|α| > 1/2$, we show that the quasi-Ablowitz-Segur and quasi-Hastings-McLeod solutions possess $[ \, |α| + \frac{1}{2} \, ]$ simple poles on the real axis, which rigorously justifies the numerical results in Fornberg and Weideman (Found. Comput. Math., 14 (2014), no. 5, 985-1016).

13 pages, 2 figures, typos corrected, references added