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

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 E min and the associated chemical potential μ 0 . On the basis of these formulae, we generalize the variational method to obtain approximate solutions, at any order of approximation, for E min , μ 0 and the ground state.