Vortices ans Polynomials: Nonuniqueness of the Adler-Moser polynomials for the Tkachenko equation

TL;DR

This paper investigates the solutions of the Tkachenko equation related to vortex equilibria, revealing the nonuniqueness of Adler-Moser polynomials and introducing a generalized equation with broader polynomial solutions.

Contribution

It demonstrates that Adler-Moser polynomials are not unique solutions to the Tkachenko equation and extends the equation to include translating equilibria with new polynomial solutions.

Findings

Adler-Moser polynomials are not unique solutions

Generalized Tkachenko equation admits non-triangular degree polynomial solutions

Stationary and translating vortex equilibria can be described by the equations

Abstract

Stationary and translating relative equilibria of point vortices in the plane are studied. It is shown that stationary equilibria of a system containing point vortices with arbitrary choice of circulations can be described with the help of the Tkachenko equation. It is obtained that the Adler - Moser polynomial are not unique polynomial solutions of the Tkachenko equation. A generalization of the Tkachenko equation to the case of translating relative equilibria is derived. It is shown that the generalization of the Tkachenko equation possesses polynomial solutions with degrees that are not triangular numbers.