Vortices and Polynomials

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 quadrupole 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 in the description of rational solutions of the fourth Painlevé equation.