Bose–Einstein condensates in optical lattices: Mathematical analysis and analytical approximate formulae

Abstract

We show that the GPE with cubic nonlinearity, as a model for describing the one-dimensional Bose–Einstein condensates loaded into a harmonically confined optical lattice, presents a set of ground states which is orbitally stable for any value of the self-interaction (attractive and repulsive) parameter and laser intensity. We also derive a new formalism which gives explicit expressions for the minimum energy Emin and the associated chemical potential μ0. On the basis of these formulae, we generalize the variational method to obtain approximate solutions, at any order of approximation, for Emin, μ0 and the ground state.