On the existence of leapfrogging pair of circular vortex filaments

We propose and analyze a system of nonlinear partial differential equations describing the motion of a pair of vortex filaments. As a preliminary analysis, we first consider the case when the filaments are arranged as straight and parallel lines, and explicitly solve the system to show that the motion of the lines resemble that of point vortices moving in a plane. Then, we consider the motion of a pair of coaxial circular vortex filaments. We show that in this case, the system can be reduced to a two-dimensional Hamiltonian system. Based on this formulation, we give a condition for the initial configuration and parameters of the filaments for leapfrogging to occur, and prove that the condition is in fact necessary andsufficient.