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We employ KAM theory to rigorously investigate quasiperiodic dynamics incigar-shaped Bose-Einstein condensates (BEC) in periodic lattices andsuperlattices. Toward this end, we apply a coherent structure ansatz to theGross-Pitaevskii equation to obtain a parametrically forced Duffing equationdescribing the spatial dynamics of the condensate. For shallow-well,intermediate-well, and deep-well potentials, we find KAM tori and Aubry-Mathersets to prove that one obtains mostly quasiperiodic dynamics for condensatewave functions of sufficiently large amplitude, where the minimal amplitudedepends on the experimentally adjustable BEC parameters. We show that thisthreshold scales with the square root of the inverse of the two-body scatteringlength, whereas the rotation number of tori above this threshold isproportional to the amplitude. As a consequence, one obtains the same dynamicalpicture for lattices of all depths, as an increase in depth essentially onlyaffects scaling in phase space. Our approach is applicable to periodicsuperlattices with an arbitrary number of rationally dependent wave numbers.

Quasiperiodic Dynamics in Bose-Einstein Condensates in Periodic Lattices and Superlattices