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Large-degree asymptotics of rational Painlevé-II functions: critical behaviour

This paper is a continuation of our analysis, begun in Buckingham and Miller (2014 Nonlinearity 27 2489–577), 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 behaviour. 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.