Propagating Magnetic Vortex Rings in Ferromagnets

It is shown that bulk ferromagnets support propagating nonlinear modes that are analogous to the vortex rings, or “smoke rings,” of fluid dynamics. These are circular loops of magnetic vorticity which travel at constant velocity parallel to their axis of symmetry. The topological structure of the continuum theory has important consequences for the properties of these magnetic vortex rings. One finds that there exists a sequence of magnetic vortex rings that are distinguished by a topological invariant (the Hopf invariant). We present analytical and numerical results for the energies, velocities, and structures of propagating magnetic vortex rings in ferromagnetic materials.