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

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journal

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Detail(s)

Original languageEnglish
Article number1840004
Journal / PublicationRandom Matrices: Theory and Application
Volume7
Issue number4
Online published28 Feb 2018
Publication statusPublished - Oct 2018

Abstract

We consider the quasi-Ablowitz–Segur and quasi-Hastings–McLeod solutions of the inhomogeneous Painlevé II equation 
u′′(x) = 2u3(x) + xu(x) − α for α ∈ and ∣α∣ >1/2. 
These solutions are obtained from the classical Ablowitz–Segur (AS) and Hastings–McLeod (HM) solutions via the Bäcklund transformation, and satisfy the same asymptotic behaviors when x → ±∞. For |α| > 1/2, we show that the quasi-Ablowitz–Segur (qAS) and quasi-Hastings–McLeod (qHM) solutions possess [∣α∣ + 1/2] simple poles on the real axis, which rigorously justifies the numerical results in Fornberg and Weideman (A computational exploration of the second Painlevé equation, Found. Comput. Math. 14(5) (2014) 985–1016).

Research Area(s)

  • Ablowitz–Segur solutions, Bäcklund transformation, Hastings–McLeod solutions, Painlevé II equation