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

TL;DR

This paper investigates the existence, stability, and symmetry of relative equilibria in vortex systems with one dominant vortex and multiple infinitesimal vortices, extending previous models and using algebraic geometry for counting solutions.

Contribution

It generalizes the reduction method for the (1+N)-vortex problem to include vortices with varying signs and weights, and characterizes symmetric and asymmetric equilibria.

Findings

Symmetric configurations require equal circulation parameters in the (1+3)-vortex problem.

Existence of stable asymmetric relative equilibria.

Use of algebraic geometry to count all relative equilibria.

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.