Axially symmetric soliton solutions in a Skyrme–Faddeev-type model with Gies's extension

Abstract

We construct static soliton solutions with non-zero Hopf topological charges to a theory which is the extended Skyrme–Faddeev model with a further quartic term in derivatives. We use an axially symmetric ansatz based on toroidal coordinates, and solve the resulting two coupled nonlinear partial differential equations in two variables by a successive over-relaxation method. We construct numerical solutions with the Hopf charge up to 4. The solutions present an interesting behavior under the changes of a special combination of the coupling constants of the quartic terms.