A model for Hopfions on the space–time S3×R

We construct static and time dependent exact soliton solutions for a theory of scalar fields taking values on a wide class of two dimensional target spaces, and defined on the four dimensional space–time S3×R. The construction is based on an ansatz built out of special coordinates on S3. The requirement for finite energy introduce boundary conditions that determine an infinite discrete spectrum of frequencies for the oscillating solutions. For the case where the target space is the sphere S2, we obtain static soliton solutions with nontrivial Hopf topological charges. In addition, such Hopfions can oscillate in time, preserving their topological Hopf charge, with any of the frequencies belonging to that infinite discrete spectrum.

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