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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.