Person:
Chow, Shui-Nee

Permanent Link
https://hdl.handle.net/1853/71155
Associated Organization(s)
Organizational Unit
ORCID
ArchiveSpace Name Record

Filters

2000 - 2005 4
4
4
4
4
4
Reset filters

Settings
End date: 2005 × Start date: 2000 × search.filters.applied.f.isOrgUnitOfPublicationTitle: School of Mathematics ×

Publication Search Results

Now showing 1 - 4 of 4
  • Thumbnail Image
    Item
    Quasiperiodic and Chaotic Dynamics in Bose-Einstein Condensates in Periodic Lattices and Superlattices
    (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.
  • Thumbnail Image
    Item
    Quasiperiodic Dynamics in Hamiltonian 1 1/2 Degree of Freedom Systems Far from Integrability
    (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.
  • Thumbnail Image
    Item
    Travelling Wave Solutions in a Tissue Interaction Model for Skin Pattern Formation
    (Georgia Institute of Technology, 2003) Ai, Shangbing ; Chow, Shui-Nee ; Yi, Yingfei
    We discuss the existence and the uniqueness of travelling wave solutions for a tissue interaction model on skin pattern formation proposed by Cruywagen and Murray. The geometric theory of singular perturbations is employed.
  • Thumbnail Image
    Item
    The Cyclicity of Period Annulus of Degenerate Quadratic Hamiltonian System with Elliptic Segment
    (Georgia Institute of Technology, 2000) Chow, Shui-Nee ; Li, Chengzhi ; Yi, Yingfei
    We study the cyclicity of period annuli (or annulus) for general degenerate quadratic Hamiltonian systems with an elliptic segment or a saddle loop, under quadratic perturbations. By using geometrical arguments and studying the respective Abelian integral based on the Picard-Fuchs equation, it is shown that the cyclicity of period annuli or annulus for such systems equals two. This result, together with those of [8],[10],[11],[18],[19], gives a complete solution to the infinitesimal Hilbert 16th problem in the case of degenerate quadratic Hamiltonian systems under quadratic perturbations.