Large-Degree Asymptotics of Rational Painlevé-IV Functions Associated to Generalized Hermite Polynomials

Abstract

Abstract The Painlevé-IV equation has three families of rational solutions generated by the generalized Hermite polynomials. Each family is indexed by two positive integers $m$ and $n$. These functions have applications to nonlinear wave equations, random matrices, fluid dynamics, and quantum mechanics. Numerical studies suggest the zeros and poles form a deformed $n\times m$ rectangular grid. Properly scaled, the zeros and poles appear to densely fill certain curvilinear rectangles as $m,n\to \infty $ with $r:=m/n$ a fixed positive real number. Generalizing a method of Bertola and Bothner [2] used to study rational Painlevé-II functions, we express the generalized Hermite rational Painlevé-IV functions in terms of certain non-Hermitian orthogonal polynomials. Using the Deift–Zhou nonlinear steepest-descent method, we asymptotically analyze the associated Riemann–Hilbert problem in the limit $n\to \infty $ with $m=r\cdot n$ for $r$ fixed. We obtain an explicit characterization of the boundary curve and determine the leading-order asymptotic expansion of the rational Painlevé-IV functions associated to generalized Hermite polynomials in the pole-free region.