Mathematics

In this talk we consider the one-dimensional non-linear Schrodinger equation with a periodic potential. In the semiclassical limit it turns out that the stationary solution are associated to the solution of a suitable discrete non-linear equation. In particular, when the strength of the non-linearity is large enough these stationary solutions are localised on a single lattice cell of the periodic potential. Such an effect may be associated to the phase transition effect observed in the Bose-Hubbard model. We plan to conclude the talk considering the case when an external Stark-type perturbation is added.