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This paper has been published in Nonlinearity. To download a preprint of this paper just click here.

This material is based upon work supported by the National Science Foundation under Grant Nos. DMS-0807653 and DMS-1206131. Any opinions, findings and conclusions or recomendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation (NSF).

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Large-Degree Asymptotics of Rational Painlevé-II Functions. Critical behavior.

Robert J. Buckingham and Peter D. Miller

Department of Mathematical Sciences, University of Cincinnati
Department of Mathematics, University of Michigan, Ann Arbor


This paper is a continuation of our analysis, begun in [J45], of the rational solutions of the inhomogeneous Painlevé-II equation and associated rational solutions of the homogeneous coupled Painlevé-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 Painlevé-II functions and also a degeneration to the tritronquée solution of the Painlevé-I equation. Our rigorous analysis is based on the steepest descent method applied to a Riemann-Hilbert representation of the rational Painlevé-II functions, and supplies leading-order formulae as well as error estimates.