This book provides a comprehensive treatment of the Gross Pitaevskii equation with a periodic potential; in particular, the localized modes supported by the periodic potential. It takes the mean-field model of the Bose Einstein condensation as the starting point of analysis and addresses the existence and stability of localized modes. The mean-field model is simplified further to the coupled nonlinear Schrodinger equations, the nonlinear Dirac equations, and the discrete nonlinear Schrodinger equations. One of the important features of such systems is the existence of band gaps in the wave...