Large-degree asymptotics of rational Painleve-IV solutions by the isomonodromy method

TL;DR

This paper rigorously analyzes the asymptotic behavior of rational solutions to the Painleve-IV equation with large parameters using the isomonodromy and steepest-descent methods, revealing algebraic and elliptic approximations.

Contribution

It introduces a Riemann-Hilbert problem framework for all rational solutions and provides rigorous asymptotic descriptions in the large-parameter limit.

Findings

Elliptic approximation valid on a curvilinear rectangle and triangles.

Algebraic approximation valid in the unbounded domain.

Excellent agreement between theoretical predictions and numerical pole-zero plots.

Abstract

The Painleve-IV equation has two families of rational solutions generated respectively by the generalized Hermite polynomials and the generalized Okamoto polynomials. We apply the isomonodromy method to represent all of these rational solutions by means of two related Riemann-Hilbert problems, each of which involves two integer-valued parameters related to the two parameters in the Painleve-IV equation. We then use the steepest-descent method to analyze the rational solutions in the limit that at least one of the parameters is large. Our analysis provides rigorous justification for formal asymptotic arguments that suggest that in general solutions of Painleve-IV with large parameters behave either as an algebraic function or an elliptic function. Moreover, the results show that the elliptic approximation holds on the union of a curvilinear rectangle and, in the case of the generalized Okamoto rational solutions, four curvilinear triangles each of which shares an edge with the rectangle; the algebraic approximation is valid in the complementary unbounded domain. We compare the theoretical predictions for the locations of the poles and zeros with numerical plots of the actual poles and zeros obtained from the generating polynomials, and find excellent agreement.