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

TL;DR

This paper derives asymptotic formulas for rational solutions of the Painleve-II equation in large-degree limits, revealing their behavior near a specific complex plane region and connecting to Painleve-I solutions.

Contribution

It extends previous analysis by providing rigorous asymptotic formulas near a critical region using Riemann-Hilbert techniques.

Findings

Asymptotic formulas valid near a specific complex region

Degeneration to trigonometric functions and Painleve-I tritronquee solution

Error estimates for the asymptotic approximations

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.