In the first part of this thesis, the dynamics of interacting ultracold bosonic atomic gases trapped in an optical lattice are examined from the perspective of nonlinear band theory. The mean-field Gross-Pitaevskii equation is used to model the Bloch waves for weakly and strongly interacting gases with a Kronig-Penney potential, i.e. a lattice of delta functions. The appearance of looped swallowtail structures in the energy bands is a highly nonlinear effect. These swallowtails are then related to period-doubled Bloch states by examining a two-color lattice. A stability analysis shows that the effective mass of the atoms is a main feature describing the stability properties of the system.

}, year = {2008}, publisher = {University of Colorado Boulder}, address = {Boulder}, }