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.
- Bose-Einstein condensates in periodic lattices
- Nonlinear Schrödinger and Discrete Nonlinear Schrödinger equations
- Semiclassical approximation
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics