5,250 research outputs found
MATHEMATICAL THEORY AND NUMERICAL METHODS FOR BOSE-EINSTEIN CONDENSATION WITH HIGHER ORDER INTERACTIONS
Ph.DDOCTOR OF PHILOSOPH
A Regularized Newton Method for Computing Ground States of Bose-Einstein condensates
In this paper, we propose a regularized Newton method for computing ground
states of Bose-Einstein condensates (BECs), which can be formulated as an
energy minimization problem with a spherical constraint. The energy functional
and constraint are discretized by either the finite difference, or sine or
Fourier pseudospectral discretization schemes and thus the original infinite
dimensional nonconvex minimization problem is approximated by a finite
dimensional constrained nonconvex minimization problem. Then an initial
solution is first constructed by using a feasible gradient type method, which
is an explicit scheme and maintains the spherical constraint automatically. To
accelerate the convergence of the gradient type method, we approximate the
energy functional by its second-order Taylor expansion with a regularized term
at each Newton iteration and adopt a cascadic multigrid technique for selecting
initial data. It leads to a standard trust-region subproblem and we solve it
again by the feasible gradient type method. The convergence of the regularized
Newton method is established by adjusting the regularization parameter as the
standard trust-region strategy. Extensive numerical experiments on challenging
examples, including a BEC in three dimensions with an optical lattice potential
and rotating BECs in two dimensions with rapid rotation and strongly repulsive
interaction, show that our method is efficient, accurate and robust.Comment: 25 pages, 6 figure
Modulated Amplitude Waves in Bose-Einstein Condensates
We analyze spatio-temporal structures in the Gross-Pitaevskii equation to
study the dynamics of quasi-one-dimensional Bose-Einstein condensates (BECs)
with mean-field interactions. A coherent structure ansatz yields a
parametrically forced nonlinear oscillator, to which we apply Lindstedt's
method and multiple-scale perturbation theory to determine the dependence of
the intensity of periodic orbits (``modulated amplitude waves'') on their wave
number. We explore BEC band structure in detail using Hamiltonian perturbation
theory and supporting numerical simulations.Comment: 5 pages, 4 figs, revtex, final form of paper, to appear in PRE
(forgot to include \bibliography command in last update, so this is a
correction of that; the bibliography is hence present again
A Perturbative Analysis of Modulated Amplitude Waves in Bose-Einstein Condensates
We apply Lindstedt's method and multiple scale perturbation theory to analyze
spatio-temporal structures in nonlinear Schr\"odinger equations and thereby
study the dynamics of quasi-one-dimensional Bose-Einstein condensates with
mean-field interactions. We determine the dependence of the intensity of
modulated amplitude waves on their wave number. We also explore the band
structure of Bose-Einstein condensates in detail using Hamiltonian perturbation
theory and supporting numerical simulations.Comment: 24 pages, 20 figs (numbered to 9), 6 tables, to appear in Chao
Accurate and efficient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates via the nonuniform FFT
In this paper, we propose efficient and accurate numerical methods for
computing the ground state and dynamics of the dipolar Bose-Einstein
condensates utilising a newly developed dipole-dipole interaction (DDI) solver
that is implemented with the non-uniform fast Fourier transform (NUFFT)
algorithm. We begin with the three-dimensional (3D) Gross-Pitaevskii equation
(GPE) with a DDI term and present the corresponding two-dimensional (2D) model
under a strongly anisotropic confining potential. Different from existing
methods, the NUFFT based DDI solver removes the singularity by adopting the
spherical/polar coordinates in Fourier space in 3D/2D, respectively, thus it
can achieve spectral accuracy in space and simultaneously maintain high
efficiency by making full use of FFT and NUFFT whenever it is necessary and/or
needed. Then, we incorporate this solver into existing successful methods for
computing the ground state and dynamics of GPE with a DDI for dipolar BEC.
Extensive numerical comparisons with existing methods are carried out for
computing the DDI, ground states and dynamics of the dipolar BEC. Numerical
results show that our new methods outperform existing methods in terms of both
accuracy and efficiency.Comment: 26 pages, 5 figure
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