Large-degree asymptotics of rational Painlevé-II functions: noncritical behaviour (Q2925982)

It is known that the poles of the rational solutions to the second Painlevé equation PII form a regular structure that fills a triangular domain in the complex plane of the independent variable. Furthermore, in the interior of this domain, the large-degree rational solutions to PII can be approximated using a modulated elliptic function which degenerates on the boundary of this triangular domain. In the present paper, the authors refine these old results performing a rigorous asymptotic analysis of the Riemann-Hilbert problem associated with the Garnier-Jimbo-Miwa isomonodromy system for PII. They compute the leading order asymptotics of the large-degree rational solutions and some auxiliary functions for all values of the independent variable except for some neighborhood of the boundary of the above mentioned triangular domain. In the exterior of this domain, rational solutions of PII are uniformly approximated by a particular root of a cubic polynomial. For any compact subset of the interior of the triangular domain, the large-degree rational solutions to PII are uniformly approximated by explicitly constructed modulated elliptic functions.