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The Painleve-IV equation has two families of rational solutions generatedrespectively by the generalized Hermite polynomials and the generalized Okamotopolynomials. We apply the isomonodromy method to represent all of theserational solutions by means of two related Riemann-Hilbert problems, each ofwhich involves two integer-valued parameters related to the two parameters inthe Painleve-IV equation. We then use the steepest-descent method to analyzethe rational solutions in the limit that at least one of the parameters islarge. Our analysis provides rigorous justification for formal asymptoticarguments that suggest that in general solutions of Painleve-IV with largeparameters behave either as an algebraic function or an elliptic function.Moreover, the results show that the elliptic approximation holds on the unionof a curvilinear rectangle and, in the case of the generalized Okamoto rationalsolutions, four curvilinear triangles each of which shares an edge with therectangle; the algebraic approximation is valid in the complementary unboundeddomain. We compare the theoretical predictions for the locations of the polesand zeros with numerical plots of the actual poles and zeros obtained from thegenerating polynomials, and find excellent agreement.

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