  We present a theoretical description of Bogolyubov-type excitations of
exciton-polariton Bose-Einstein condensates (BECs) in semiconductor
microcavities. For a typical two dimensional (2D) BEC we focus on two limiting
cases, the weak- and strong-coupling regimes, where a perturbation theory and
the Thomas-Fermi approximation, respectively, are valid. We calculate
integrated scattering intensity spectra for probing the collective excitations
of the condensate in both considered limits. Moreover, in relation to recent
experiments on optical modulation allowing localization of condensates in a
trap with well controlled shape and dimensions, we study the quasi-one
dimensional (1D) motion of the BEC in microwires and report the corresponding
Bogolyubov's excitation spectrum. We show that in 1D case the characteristic
polariton-polariton interaction constant is expressed as $g_{1}=3\lambda
\mathcal{N}/(2L_{y})$ ($\lambda $ is the 2D polariton-polaritons interaction
parameter in the cavity, $\mathcal{N}$ the number of the particles, and $L_{y}$
the wirecavity width). We reveal some interesting features for 2D and 1D
Bogolyubov spectra for both repulsive $(\lambda >0)$ and attractive $(\lambda
<0)$ interaction.
