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We investigated the effects of the spacetime curvature and extra dimensionson the excitations of a self-interacting vector field known as the bumblebeefield. The self-interacting quadratic potential breaks the gauge invariance andthe vacuum expectation value (VEV) of the bumblebee field $b_M$ violates thelocal particle Lorentz symmetry. By assuming the bumblebee field living in a$AdS_{5}$ bulk, we found an exponential suppression of the self-interactingconstant $\lambda$ and the bumblebee VEV along the extra dimension. Thefluctuations of the bumblebee upon the VEV can be decomposed into transverseand longitudinal modes with respect to $b_{M}$. Despite the curvature, thetransverse mode acquires massive Kaluza-Klein towers, while the longitudinalmode acquires LV mass $\lambda b^{2}$. On the other hand, the currentconservation law prevents massive Kaluza-Klein modes for the longitudinal mode.For a spacelike $b_{M}$ along the extra dimension and assuming a FRW 3-braneembedded in the $AdS_{5}$ yields to an additional dissipative term to thelongitudinal mode. The cosmological expansion leads to decay of thelongitudinal mode in a time $\Delta t \approx H^{-1}$, where $H=\dot{a}/a$ isthe 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 decaysin time with $a^{-3}$ and in the extra dimension with $z^{-\lambdab^{2}l^{2}}$.

The bumblebee field excitations in a cosmological braneworld