Vortices and Polynomials

TL;DR

This paper explores the connection between point vortex dynamics and special polynomials, revealing how roots of these polynomials correspond to vortex equilibria and configurations in fluid dynamics.

Contribution

It identifies new polynomial solutions related to vortex configurations, linking classical and special polynomials to fluid dynamics equilibria.

Findings

Roots of Hermite polynomials describe equilibria of identical vortices.

Zeros of Adler-Moser polynomials correspond to stationary vortex configurations.

New solutions involve polynomials from the fourth Painleve equation.

Abstract

The relationship between point vortex dynamics and the properties of polynomials with roots at the vortex positions is discussed. Classical polynomials, such as the Hermite polynomials, have roots that describe the equilibria of identical vortices on the line. Stationary and uniformly translating vortex configurations with vortices of the same strength but positive or negative orientation are given by the zeros of the Adler-Moser polynomials, which arise in the description of rational solutions of the Korteweg-de Vries equation. For quadupole background flow, vortex configurations are given by the zeros of polynomials expressed as wronskians of Hermite polynomials. Further new solutions are found in this case using the special polynomials arising the in the description of rational solutions of the fourth Painleve equation.