We employ KAM theory to rigorously investigate quasiperiodic dynamics in
cigar-shaped Bose-Einstein condensates (BEC) in periodic lattices and
superlattices. Toward this end, we apply a coherent structure ansatz to the
Gross-Pitaevskii equation to obtain a parametrically forced Duffing equation
describing the spatial dynamics of the condensate. For shallow-well,
intermediate-well, and deep-well potentials, we find KAM tori and Aubry-Mather
sets to prove that one obtains mostly quasiperiodic dynamics for condensate
wave functions of sufficiently large amplitude, where the minimal amplitude
depends on the experimentally adjustable BEC parameters. We show that this
threshold scales with the square root of the inverse of the two-body scattering
length, whereas the rotation number of tori above this threshold is
proportional to the amplitude. As a consequence, one obtains the same dynamical
picture for lattices of all depths, as an increase in depth essentially only
affects scaling in phase space. Our approach is applicable to periodic
superlattices with an arbitrary number of rationally dependent wave numbers.Comment: 29 pages, 6 figures (several with multiple parts; higher-quality
versions of some of them available at
http://www.its.caltech.edu/~mason/papers), to appear very soon in Journal of
Nonlinear Scienc