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

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

2011-12-19

arxiv.org/abs/1112.4350v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

15 pages, 2 figures

Scientific paper

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.

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