Isospinning hopfions

The problem of constructing internally rotating solitons of fixed angular frequency ω in the Faddeev-Skyrme model is reformulated as a variational problem for an energy-like functional, called pseudoenergy, which depends parametrically on ω. This problem is solved numerically using a gradient descent method, without imposing any spatial symmetries on the solitons, and the dependence of the solitons' energy on ω, and on their conserved total isospin J, studied. It is found that, generically, the shape of a soliton is independent of ω, and that its size grows monotonically with ω. A simple elastic rod model of time-dependent hopfions is developed which, despite having only one free parameter, accounts well for most of the numerical results.