  We show the existence of self-dual (topological) solitons in a gauged version
of the baby Skyrme model in which the Born-Infeld term governs the gauge field
dynamics. The successful implementation of the Bogomol'nyi-Prasad-Sommerfield
formalism provides a lower bound for the energy and the respective self-dual
equations whose solutions are the solitons saturating such a limit. The energy
lower bound (Bogomol'nyi bound) is proportional to the topological charge of
the Skyrme field and therefore quantized. In contrast, the total magnetic flux
is a nonquantized quantity. Furthermore, the model supports three types of
self-dual solitons profiles: the first describes compacton solitons, the second
follows a Gaussian decay law, and the third portrays a power-law decay.
Finally, we perform numerical solutions of the self-dual equations and depicted
the soliton profiles for different values of the parameters controlling the
nonlinearity of the model.
