Existence, Stability, and Symmetry of Relative Equilibria with a Dominant Vortex

Source 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.