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

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.

15 pages, 2 figures

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• Vortices and Polynomials
• Equilibrium of charges and differential equations solved by polynomials
• Point vortices and polynomials of the Sawada-Kotera and Kaup-Kupershmidt equations
• Integrable Dynamics of Charges Related to Bilinear Hypergeometric Equation
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Cited by in corpus (5)

• Generating Functions, Polynomials and Vortices with Alternating Signs in Bose-Einstein Condensates
• Point vortices and classical orthogonal polynomials
• Electrostatic partners and zeros of orthogonal and multiple orthogonal polynomials
• Multi-particle dynamical systems and polynomials
• Equilibrium of Charges and Differential Equations Solved by Polynomials II