Exactly formulated Bose–Einstein condensates in optical lattices

In light of the recent experimental observation of Bose–Einstein condensates held in 1D, 2D and 3D optical lattice potentials, we derive several families of exact solutions of the Gross–Pitaevskii equation from some balance conditions between the optical potentials and interaction energies. The exact solutions in same dimension possess same norm but different phases, which describe the condensed atoms with same spatial distribution and average number of atoms per well but different transport velocities and atomic current densities.