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arXiv · 1406.0826

Large-degree asymptotics of rational Painleve-II functions. II

Robert J. Buckingham·Peter D. Miller
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Abstract

This paper is a continuation of our analysis, begun in arXiv:1310.2276, of the rational solutions of the inhomogeneous Painleve-II equation and associated rational solutions of the homogeneous coupled Painleve-II system in the limit of large degree. In this paper we establish asymptotic formulae valid near a certain curvilinear triangle in the complex plane that was previously shown to separate two distinct types of asymptotic behavior. Our results display both a trigonometric degeneration of the rational Painleve-II functions and also a degeneration to the tritronquee solution of the Painleve-I equation. Our rigorous analysis is based on the steepest descent method applied to a Riemann-Hilbert representation of the rational Painleve-II functions, and supplies leading-order formulae as well as error estimates.

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BibTeXRIS

Robert J. Buckingham, Peter D. Miller. 2014-06-02. Large-degree asymptotics of rational Painleve-II functions. II. https://doi.org/10.1088/0951-7715%2F27%2F10%2F2489

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