  In this paper, we have investigated theoretically the influence of atomic
collisions on the behaviour of a one-dimensional Bose-Einstein condensate
inside a driven optical cavity. We develop the discrete-mode approximation for
the condensate taking into account the interband transitions due to the s-wave
scattering interaction. We show that in the Bogoliubov approximation the
atom-atom interaction shifts the energies of the excited modes and also plays
the role of an optical parametric amplifier for the Bogoliubov side mode which
can affect its normal-mode splitting behaviour. On the other hand due to the
atomic collisions the resonance frequency of the cavity is shifted which leads
to the decrease of the number of cavity photons and the depletion of the
Bogoliubov mode. Besides, it reduces the effective atom-photon coupling
parameter which consequently leads to the decrease of the entanglement between
the Bogoliubov mode and the optical field.
