Relative Equilibria in the Four-Vortex Problem with Two Pairs of Equal Vorticities

TL;DR

This paper classifies all relative equilibria configurations in the four-vortex problem with two pairs of equal vorticities, revealing symmetry properties, existence conditions, and bifurcations using algebraic geometry and analysis.

Contribution

It provides a complete classification of solutions for the four-vortex problem with two equal vorticity pairs, including symmetry, existence, and bifurcation analysis.

Findings

Convex configurations with two equal vorticity pairs are symmetric for m > 0.

Rhombus solutions exist for all m, isosceles trapezoids only for m > 0.

Asymmetric convex configurations occur when m < 0.

Abstract

We examine in detail the relative equilibria in the four-vortex problem where two pairs of vortices have equal strength, that is, \Gamma_1 = \Gamma_2 = 1 and \Gamma_3 = \Gamma_4 = m where m is a nonzero real parameter. One main result is that for m > 0, the convex configurations all contain a line of symmetry, forming a rhombus or an isosceles trapezoid. The rhombus solutions exist for all m but the isosceles trapezoid case exists only when m is positive. In fact, there exist asymmetric convex configurations when m < 0. In contrast to the Newtonian four-body problem with two equal pairs of masses, where the symmetry of all convex central configurations is unproven, the equations in the vortex case are easier to handle, allowing for a complete classification of all solutions. Precise counts on the number and type of solutions (equivalence classes) for different values of m, as well as a description of some of the bifurcations that occur, are provided. Our techniques involve a combination of analysis and modern and computational algebraic geometry.