Large-Degree Asymptotics of Rational Painlevé-IV Solutions by the Isomonodromy Method

The Painlevé-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 Painlevé-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 Painlevé-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.