(Georgia Institute of Technology, 2005-07-23)
van Noort, Martijn; Porter, Mason; Yi, Yingfei; Chow, Shui-Nee
We employ KAM theory to rigorously investigate the transition between
quasiperiodic and chaotic 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 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 its amplitude essentially only affects scaling in phase space. Our approach is applicable to periodic superlattices with an arbitrary number of rationally dependent wave numbers.
(Georgia Institute of Technology, 2005-01-21)
Chow, Shui-Nee; van Noort, Martijn; Yi, Yingfei
The subject of this paper is two-quasiperiodicity in a large class of one-and-a-half degree of freedom Hamiltonian systems. The main result is that
such systems have invariant tori for any internal frequency that is of constant
type and sufficiently large, relative to the forcing frequency. An explicit bound
on the minimum value of the internal frequency is presented. The systems
under consideration are not required to be small perturbations of integrable
ones.