Relative equilibria in the four-vortex problem with two pairs of equal vorticities

We examine in detail the relative equilibria in the planar four-vortex problem where two pairs of vortices have equal strength, that is, Γ1 = Γ2 = 1 and Γ3 = Γ4 = m where m R-{0} is a parameter. One main result is that, form>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.