Connection formulas for the Ablowitz–Segur solutions of the inhomogeneous Painlevé II equation

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

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Original languageEnglish
Pages (from-to)2982-3009
Journal / PublicationNonlinearity
Issue number7
Publication statusPublished - 19 Jun 2017


We consider the second Painlevé equation u''(x)=2u3(x)+xu(x)−α, where α is a nonzero constant. Using the Deift–Zhou nonlinear steepest descent method for Riemann–Hilbert problems, we rigorously prove the asymptotics as x →±∞ 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 α ∈ (-1/2, 1/2) .

Research Area(s)

  • connection formulas, Painlevé II equation, Riemann-Hilbert problem