  We studied a system of atomic Bose-Einstein condensate coupled to a ring
cavity within the mean-field theory. Due to the interaction between atoms and
light field, the atoms can be self-trapped. This is verified with both
variational and numerical methods. We examined the stability of these
self-trapped states. For a weakly pumped cavity, they spread during the
evolution; while at strong pumping, they can maintain the shape for a long
time. We also studied the moving dynamics of these self-trapped waves, and
found out that it can be strongly affected by the cavity decay rate. For a
small cavity decay rate, the self-trapped waves undergo a damped oscillation.
Increasing the cavity decay rate will lead to a deceleration of the
self-trapped waves. We also compared the main results with the semiclassical
theory in which atoms are treated as classical particles.
