1 February 2017 Transition asymptotics for the Painlevé II transcendent
Thomas Bothner
Duke Math. J. 166(2): 205-324 (1 February 2017). DOI: 10.1215/00127094-3714650

Abstract

We consider real-valued solutions u=u(x|s), xR, of the second Painlevé equation uxx=xu+2u3 which are parameterized in terms of the monodromy data s(s1,s2,s3)C3 of the associated Flaschka–Newell system of rational differential equations. Our analysis describes the transition, as x, between the oscillatory power-like decay asymptotics for |s1|<1 (Ablowitz–Segur) to the power-like growth behavior for |s1|=1 (Hastings–McLeod) and from the latter to the singular oscillatory power-like growth for |s1|>1 (Kapaev). It is shown that the transition asymptotics are of Boutroux type; that is, they are expressed in terms of Jacobi elliptic functions. As applications of our results we obtain asymptotics for the Airy kernel determinant det(IγKAi)|L2(x,) in a double scaling limit x, γ1, as well as asymptotics for the spectrum of KAi.

Citation

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Thomas Bothner. "Transition asymptotics for the Painlevé II transcendent." Duke Math. J. 166 (2) 205 - 324, 1 February 2017. https://doi.org/10.1215/00127094-3714650

Information

Received: 5 March 2015; Revised: 8 February 2016; Published: 1 February 2017
First available in Project Euclid: 16 December 2016

zbMATH: 1369.34107
MathSciNet: MR3600752
Digital Object Identifier: 10.1215/00127094-3714650

Subjects:
Primary: 33E17
Secondary: 33C10 , 34E05 , 34M50

Keywords: Deift–Zhou nonlinear steepest descent method , Riemann–Hilbert problem , second Painlevé equation , transition asymptotics

Rights: Copyright © 2017 Duke University Press

Vol.166 • No. 2 • 1 February 2017
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