  On examples of Bose-Einstein condensates embedded in two-dimensional optical
lattices we show that in nonlinear periodic systems modulational instability
and inter-band tunneling are intrinsically related phenomena. By direct
numerical simulations we found that tunneling results in attenuation or
enhancement of instability. On the other hand, instability results in
asymmetric nonlinear tunneling. The effect strongly depends on the band gap
structure and it is especially significant in the case of the resonant
tunneling. The symmetry of the coherent structures emerging from the
instability reflects the symmetry of both the stable and the unstable states
between which the tunneling occurs. Our results provide an evidence of profound
effect of the band structure on superfluid-insulator transition.
