Two-species Bose-Einstein condensate in an optical lattice: analytical approximate formul\ae

Employing a general variational method and perturbation theory, we derived explicit solutions for the description of one-dimensional two species Bose-Einstein condensates confined by a harmonic trap potential in an optical lattice. We consider the system of two coupled Gross-Pitaevkii equations (GPE) and derive explicit expressions for the chemical potentials and wavefunctions in terms of the atom-atom interaction parameters and laser intensity. We have compared our results with the numerical solutions of the GPE and performed a quantitative analysis for the both considered methods. We underline the importance of the obtained explicit solutions to characterize the density profile or degree of miscibility of the two components.