Comparisons between classical and quantum mechanical nonlinear lattice models

Abstract: In the mid-1920s, the great Albert Einstein proposed that at extremely low temperatures, a gas of bosonic particles will enter a new phase where a large fraction of them occupy the same quantum state. This state would bring many of the peculiar features of quantum mechanics, previously reserved for small samples consisting only of a few atoms or molecules, up to a macroscopic scale. This is what we today call a Bose-Einstein condensate. It would take physicists almost 70 years to realize Einstein's idea, but in 1995 this was finally achieved.The research on Bose-Einstein condensates has since taken many directions, one of the most exciting being to study their behavior when they are placed in optical lattices generated by laser beams. This has already produced a number of fascinating results, but it has also proven to be an ideal test-ground for predictions from certain nonlinear lattice models.Because on the other hand, nonlinear science, the study of generic nonlinear phenomena, has in the last half century grown out to a research field in its own right, influencing almost all areas of science and physics. Nonlinear localization is one of these phenomena, where localized structures, such as solitons and discrete breathers, can appear even in translationally invariant systems. Another one is the (in)famous chaos, where deterministic systems can be so sensitive to perturbations that they in practice become completely unpredictable. Related to this is the study of different types of instabilities; what their behavior are and how they arise.In this thesis we compare classical and quantum mechanical nonlinear lattice models which can be applied to BECs in optical lattices, and also examine how classical nonlinear concepts, such as localization, chaos and instabilities, can be transfered to the quantum world.