  We investigated the effects of the spacetime curvature and extra dimensions
on the excitations of a self-interacting vector field known as the bumblebee
field. The self-interacting quadratic potential breaks the gauge invariance and
the vacuum expectation value (VEV) of the bumblebee field $b_M$ violates the
local particle Lorentz symmetry. By assuming the bumblebee field living in a
$AdS_{5}$ bulk, we found an exponential suppression of the self-interacting
constant $\lambda$ and the bumblebee VEV along the extra dimension. The
fluctuations of the bumblebee upon the VEV can be decomposed into transverse
and longitudinal modes with respect to $b_{M}$. Despite the curvature, the
transverse mode acquires massive Kaluza-Klein towers, while the longitudinal
mode acquires LV mass $\lambda b^{2}$. On the other hand, the current
conservation law prevents massive Kaluza-Klein modes for the longitudinal mode.
For a spacelike $b_{M}$ along the extra dimension and assuming a FRW 3-brane
embedded in the $AdS_{5}$ yields to an additional dissipative term to the
longitudinal mode. The cosmological expansion leads to decay of the
longitudinal mode in a time $\Delta t \approx H^{-1}$, where $H=\dot{a}/a$ is
the Hubble parameter and $a(t)$ is the scale factor. For a timelike $b_{M}$,
the longitudinal mode does not propagate on the brane and its amplitude decays
in time with $a^{-3}$ and in the extra dimension with $z^{-\lambda
b^{2}l^{2}}$.
