Abstract: We study anharmonic (optical or magnetic) periodic potentials and demonstrate that they may exhibit unusual features in the band-gap spectra and the nature of the matter-wave Bloch waves and solitons. We reveal that the band gaps may be strongly modified and even close, and show that changing the potential shape may change the symmetry of the gap-edge Bloch waves and affect the symmetry of higher-order gap solitons. We introduce a mixed-gap matter-wave interface soliton which belongs simultaneously to two different spectral gaps and describe interface localized states in chirped periodic potentials. Finally, we generalize our results to two-dimensional periodic potentials and examine the band-gap structure, matter-wave gap solitons, and lattice vortex states.

Keywords: magnetic lattices; ultracold atoms; einstein

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