  We analyze stability of superfluid currents in a system of strongly
interacting ultra-cold atoms in an optical lattice. We show that such a system
undergoes a dynamic, irreversible phase transition at a critical phase gradient
that depends on the interaction strength between atoms. At commensurate
filling, the phase boundary continuously interpolates between the classical
modulation instability of a weakly interacting condensate and the equilibrium
quantum phase transition into a Mott insulator state at which the critical
current vanishes. We argue that quantum fluctuations smear the transition
boundary in low dimensional systems. Finally we discuss the implications to
realistic experiments.
