  We have shown the existence of self-dual solitons in a type of generalized
Chern-Simons baby Skyrme model where the generalized function (depending only
in the Skyrme field) is coupled to the sigma-model term. The consistent
implementation of the Bogomol'nyi-Prasad-Sommerfield (BPS) formalism requires
the generalizing function becomes the superpotential defining properly the
self-dual potential. Thus, we have obtained a topological energy lower-bound
(Bogomol'nyi bound) and the self-dual equations satisfied by the fields
saturating such a bound. The Bogomol'nyi bound being proportional to the
topological charge of the Skyrme field is quantized whereas the total magnetic
flux is not. Such as expected in a Chern-Simons model the total magnetic flux
and the total electrical charge are proportional to each other. Thus, by
considering the superpotential a well-behaved function in the whole target
space we have shown the existence of three types of self-dual solutions:
compacton solitons, soliton solutions whose tail decays following an
exponential-law $e^{-\alpha r^{2}}$ ($\alpha>0$), and solitons having a
power-law decay $r^{-\beta}$ ($\beta>0$). The profiles of the two last solitons
can exhibit a compactonlike behavior. The self-dual equations have been solved
numerically and we have depicted the soliton profiles, commenting on the main
characteristics exhibited by them.
