  The criteria for validity of adiabaticity for nonlinear wave equations are
considered within the context of atomic matter-waves tunneling from
macroscopically populated optical standing-wave traps loaded from a
Bose-Einstein condensate. We show that even when the optical standing wave is
slowly turned on and the condensate behaves adiabatically during this turn-on,
once the tunneling-time between wells in the optical lattice becomes longer
than the nonlinear time-scale, adiabaticity breaks down and a significant
spatially varying phase develops across the condensate wave function from well
to well. This phase drastically affects the contrast of the fringe pattern in
Josephson-effect interference experiments, and the condensate coherence
properties in general.
