  We present our experimental investigations on the subject of dynamical
nonlinearity-induced instabilities and of nonlinear Landau-Zener tunneling
between two energy bands in a Rubidium Bose-Einstein condensate in an
accelerated periodic potential. These two effects may be considered two
different regimes (for small and large acceleration) of the same physical
system and studied with the same experimental protocol. Nonlinearity introduces
an asymmetry in Landau-Zener tunneling; as a result, tunneling from the ground
state to the excited state is enhanced whereas in the opposite direction it is
suppressed. When the acceleration is lowered, the condensate exhibits an
unstable behaviour due to nonlinearity. We also carried out a full numerical
simulation of both regimes integrating the full Gross-Pitaevskii equation; for
the Landau-Zener effect we also used a simple two-level model. In both cases we
found good agreement with the experimental results.
