Vortex rings in ferromagnets: Numerical simulations of the time-dependent three-dimensional Landau-Lifshitz equation

Vortex ring solutions are presented for the Landau-Lifshitz equation, which models the dynamics of a three-dimensional ferromagnet. The vortex rings propagate at constant speed along their symmetry axis and are characterized by the integer-valued Hopf charge. They are stable to axial perturbations, but it is demonstrated that an easy axis anisotropy results in an instability to perturbations, which breaks the axial symmetry. The unstable mode corresponds to a migration of spin flips around the vortex ring that leads to a pinching instability and, ultimately, the collapse of the vortex ring. It is found that this instability does not exist for an isotropic ferromagnet. Similarities between vortex rings in ferromagnets and vortons in cosmology are noted.