Existence, stability, and symmetry of relative equilibria with a dominant vortex

Abstract

We analyze existence, stability, and symmetry of point vortex relative equilibria with one dominant vortex and N vortices with infinitesimal circulation. The dimension of the problem can be reduced by taking an infinitesimal circulation limit, resulting in the so-called (1+N)-vortex problem. In this work, we first generalize the reduction to allow for circulations of varying signs and weights. We then prove that symmetric configurations require equality of two circulation parameters in the (1+3)-vortex problem and show that there are stable asymmetric relative equilibria. In a number of examples, we use rigorous methods from algebraic geometry to count all relative equilibria.

References in corpus (6)

• Guiding-center dynamics of vortex dipoles in Bose-Einstein condensates
• Dynamics of Vortex Dipoles in Confined Bose-Einstein Condensates
• Some Problems on the Classical N-Body Problem
• Generating Functions, Polynomials and Vortices with Alternating Signs in Bose-Einstein Condensates
• Relative Equilibria in the Four-Vortex Problem with Two Pairs of Equal Vorticities
• Relative Equilibria of the $(1+N)$-Vortex Problem

Cited by in corpus (1)

• Nonlinear Stability of Relative Equilibria in the Planar $N$-Vortex Problem with Non-Zero Total Circulation