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.
