  We have shown the existence of self-dual solutions in new Maxwell-Higgs
scenarios where the gauge field possesses a $k$-generalized dynamic, i.e., the
kinetic term of gauge field is a highly nonlinear function of
$F_{\mu\nu}F^{\mu\nu}$. We have implemented our proposal by means of a
$k$-generalized model displaying the spontaneous symmetry breaking phenomenon.
We implement consistently the Bogomol'nyi-Prasad-Sommerfield formalism
providing highly nonlinear self-dual equations whose solutions are electrically
neutral possessing total energy proportional to the magnetic flux. Among the
infinite set of possible configurations, we have found families of
$k$-generalized models whose self-dual equations have a form mathematically
similar to the ones arising in the Maxwell-Higgs or Chern-Simons-Higgs models.
Furthermore, we have verified that our proposal also supports infinite twinlike
models with $|\phi|^4$-potential or $|\phi |^6$-potential. With the aim to show
explicitly that the BPS equations are able to provide well-behaved
configurations, we have considered a test model in order to study axially
symmetric vortices. By depending of the self-dual potential, we have shown that
the $k$-generalized model is able to produce solutions that for long distances
have a exponential decay (as Abrikosov-Nielsen-Olesen vortices) or have a
power-law decay (characterizing delocalized vortices). In all cases, we observe
that the generalization modifies the vortex core size, the magnetic field
amplitude and the bosonic masses but the total energy remains proportional to
the quantized magnetic flux.
