Restricted baby Skyrme-Maxwell theory in a magnetic medium: BPS configurations and some properties

We study the existence of BPS configurations in a restricted baby Skyrme-Maxwell enlarged via the inclusion of a nontrivial magnetic permeability. In order to attain such a goal, we use the Bogomol'nyi-Prasad-Sommerfield prescription, which allows us to obtain the lower bound for the energy and the BPS equations whose [electrically neutral] solutions saturate that bound. During the energy minimization procedure, we find a differential constraint which involves the self-dual potential, the superpotential itself and also the magnetic permeability. In order to solve the BPS system, we focus our attention on those solutions with rotational symmetry. For that, we fix the magnetic permeability and select two BPS potentials which exhibit a similar behavior near to the vacuum. We depict the resulting profiles and proceed to an analytical description of the properties of the BPS magnetic field. Furthermore, we consider some essential aspects of our model, such as the conditions for the overall existence of the BPS solutions, and how the permeability affects the magnetic flux. Finally, we present a family of exact BPS solutions.