In this chapter we review the effects of spatial localization of a Bose-Einstein condensate (BEC) that arise due to the combination of the intrinsic nonlinearity of a condensate due to repulsive atomic interactions and Bragg scattering of a matter-wave on a periodic potential of an optical lattice (OL). It goes without saying that we will not discuss the trivial case of very deep periodic potentials which leads to trapping of the atoms within single sites. Instead, we will address the situation where the single particle tunneling rate is still much faster than the observation time. Under such conditions, a nontrivial localization becomes possible due to the fact that, at the edges of a Brillouin zone (BZ) of the lattice, the condensate experiences anomalous diffraction (dispersion), the magnitude of which can be controlled by tuning the depth of the OL potential. Keeping a wave packet from spreading can therefore be either achieved by actively controlling the dispersion or by utilizing the interaction between atoms. The first approach is known as dispersion/diffraction management and the second one leads to nonlinearly localized states.
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Ostrovskaya, E.A., Oberthaler, M.K., Kivshar, Y.S. (2008). Nonlinear Localization of BECs in Optical Lattices. In: Kevrekidis, P.G., Frantzeskakis, D.J., Carretero-González, R. (eds) Emergent Nonlinear Phenomena in Bose-Einstein Condensates. Atomic, Optical, and Plasma Physics, vol 45. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73591-5_6
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