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We construct the first analytic examples of topologically non-trivialsolutions of the (3+1)-dimensional $U(1)$ gauged Skyrme model within a finitebox in (3+1)-dimensional flat space-time. There are two types of gaugedsolitons. The first type corresponds to gauged Skyrmions living within a finitevolume. The second corresponds to gauged time-crystals (smooth solutions of the$U(1)$ gauged Skyrme model whose periodic time-dependence is protected by awinding number). The notion of electromagnetic duality can be extended forthese two types of configurations in the sense that the electric and one of themagnetic components can be interchanged. These analytic solutions show veryexplicitly the Callan-Witten mechanism (according to which magnetic monopolesmay "swallow" part of the topological charge of the Skyrmion) since theelectromagnetic field contribute directly to the conserved topological chargeof the gauged Skyrmions. As it happens in superconductors, the magnetic fieldis suppressed in the core of the gauged Skyrmions. On the other hand, theelectric field is strongly suppresed in the core of gauged time crystals.

Analytic topologically non-trivial solutions of the (3+1)-dimensional $U(1)$ gauged Skyrme model and extended duality