  We investigate the nonlinear dynamics of a combined system which is composed
of a cigar-shaped Bose-Einstein condensate and an optical cavity. The two sides
couple dispersively. This system is characterized by its nonlinearity: after
integrating out the freedom of the cavity mode, the potential felt by the
condensate depends on the condensate itself. We develop a discrete-mode
approximation for the condensate. Based on this approximation, we map out the
steady configurations of the system. It is found that due to the nonlinearity
of the system, the nonlinear levels of the system can fold up in some parameter
regimes. That will lead to the breakdown of adiabaticity. Analysis of the
dynamical stability of the steady states indicates that the same level
structure also results in optical bistability.
