Bose-Einstein Condensates in Optical Lattices: Band-Gap Structure and   Solitons
  We analyze the existence and stability of spatially extended (Bloch-type) and localized states of a Bose-Einstein condensate loaded into an optical lattice. In the framework of the Gross-Pitaevskii equation with a periodic potential, we study the band-gap structure of the matter-wave spectrum in both the linear and nonlinear regimes. We demonstrate the existence of families of spatially localized matter-wave gap solitons, and analyze their stability in different band gaps, for both repulsive and attractive atomic interactions. 
Pearl J.Y. Louis, Elena A. Ostrovskaya, Craig M. Savage, and Yuri S.   Kivshar