  We find numerically that in the limit of weak atom-atom interactions a
Bose-Einstein condensate in an optical lattice may develop a pulsating
dynamical instability in which the atoms nearly periodically form a peak in the
occupation numbers of the lattice sites, and then return to the unstable
initial state. Multiple peaks behaving similarly may also occur. Simple
arguments show that the pulsating instability is a remnant of integrability,
and give a handle on the relevant physical scales.
