Stationary States for Nonlinear Schrödinger Equations with Periodic Potentials

In this paper we consider a one-dimensional non-linear Schrödinger 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 Schrödinger equation to a discrete
non-linear Schrödinger 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