On the quasi-Ablowitz-Segur and quasi-Hastings-McLeod solutions of the inhomogeneous Painlev\'{e} II equation

We consider the quasi-Ablowitz-Segur and quasi-Hastings-McLeod solutions of the inhomogeneous Painlev\'e II equation $$ u"(x)=2u^3(x)+xu(x)-\alpha \qquad \textrm{for } \alpha \in \mathbb{R} \textrm{ and } |\alpha| > \frac{1}{2}. $$ These solutions are obtained from the classical Ablowitz-Segur and Hastings-McLeod solutions via the B\"acklund transformation, and satisfy the same asymptotic behaviors when $x \to \pm \infty$. For $|\alpha| > 1/2$, we show that the quasi-Ablowitz-Segur and quasi-Hastings-McLeod solutions possess $[ \, |\alpha| + \frac{1}{2} \, ]$ 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).