  We have studied the existence of self-dual effective compact and true
compacton configurations in Abelian Higgs models with generalized dynamics. We
have named of an effective compact solution the one whose profile behavior is
very similar to the one of a compacton structure but still preserves a tail in
its asymptotic decay. In particular we have investigate the electrically
neutral configurations of the Maxwell-Higgs and Born-Infeld-Higgs models and
the electrically charged ones of the Chern-Simons-Higgs and
Maxwell-Chern-Simons-Higgs models. The generalization of the kinetic terms is
performed by means of dielectric functions in gauge and Higgs sectors. The
implementation of the BPS formalism without the need to use a specific
\textit{Ansatz} has leaded us to the explicit determination of the dielectric
function associated to the Higgs sector to be proportional to $\lambda |\phi
|^{2\lambda -2}$, $\lambda >1$. Consequently, the followed procedure allows us
to determine explicitly new families of self-dual potentials for every model.
We have also observed that for sufficiently large values of $\lambda$ every
model supports effective compact vortices. The true compacton solutions arising
for $\lambda= \infty $ are analytical. Therefore, this new self-dual structures
enhance the space of BPS solutions of the Abelian Higgs models and they
probably will imply in interesting applications in physics and mathematics.
