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 $$u"(x)=2u^3(x)+xu(x)-\alpha \text{for }\alpha \in \mathbb{R}\text{ and }|\alpha |>\frac{1}{2}.$$ 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 $|\alpha |>1/2$, 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).

13 pages, 2 figures, typos corrected, references added

References in corpus (1)

• Quasi-linear Stokes phenomenon for the Hastings-McLeod solution of the second Painlevé equation

Cited by in corpus (1)

• Open Problems for Painlevé Equations