Rational solutions of the Painlevé-IV equation with large parameters

Peter Miller (University of Michigan)

11-Jan-2022, 14:00-15:00 (2 years ago)

Abstract: The Painlevé-IV equation has two families of rational solutions, which can be represented in terms of special polynomials called generalized Hermite polynomials and generalized Okamoto polynomials, respectively. The generalized Hermite polynomials have a convenient representation in terms of Hankel determinants for a suitable weight and hence can be identified with norming constants for certain pseudo-orthogonal polynomials. This connection provides a path to the analysis of the generalized Hermite rationals when the parameters are large; however it is not known whether the generalized Okamoto polynomials have a similar representation. In this talk, we explain how the isomonodromic approach places both families of rational solutions in terms of special cases of the Riemann-Hilbert inverse monodromy problem for Painlevé-IV. This allows techniques from steepest descent theory to be used to analyze both families of rational solutions within a common analytical framework. This is joint work with Robert Buckingham.

complex variablesdynamical systems

Audience: researchers in the topic


CAvid: Complex Analysis video seminar

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