Stationary States for Nonlinear Schrodinger Equations with Periodic Potentials

In this paper we consider a one-dimensional non-linear Schrodinger equation with a periodic potential. In the semiclassical limit we prove the existence of stationary solutions by means of the reduction of the non-linear Schrodinger equation to a discrete non-linear Schrodinger equation. In particular, in the limit of large nonlinearity strength the stationary solutions turn out to be localized on a single lattice site of the periodic potential. A connection of these results with the Mott insulator phase for Bose-Einstein condensates in a one-dimensional periodic lattice is also discussed.