262 research outputs found
Quasiperiodic Dynamics in Bose-Einstein Condensates in Periodic Lattices and Superlattices
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
The non-integrability of the Zipoy-Voorhees metric
The low frequency gravitational wave detectors like eLISA/NGO will give us
the opportunity to test whether the supermassive compact objects lying at the
centers of galaxies are indeed Kerr black holes. A way to do such a test is to
compare the gravitational wave signals with templates of perturbed black hole
spacetimes, the so-called bumpy black hole spacetimes. The Zipoy-Voorhees (ZV)
spacetime (known also as the spacetime) can be included in the bumpy
black hole family, because it can be considered as a perturbation of the
Schwarzschild spacetime background. Several authors have suggested that the ZV
metric corresponds to an integrable system. Contrary to this integrability
conjecture, in the present article it is shown by numerical examples that in
general ZV belongs to the family of non-integrable systems.Comment: 10 pages, 13 figure
Some special solutions to the Hyperbolic NLS equation
The Hyperbolic Nonlinear Schrodinger equation (HypNLS) arises as a model for
the dynamics of three-dimensional narrowband deep water gravity waves. In this
study, the Petviashvili method is exploited to numerically compute bi-periodic
time-harmonic solutions of the HypNLS equation. In physical space they
represent non-localized standing waves. Non-trivial spatial patterns are
revealed and an attempt is made to describe them using symbolic dynamics and
the language of substitutions. Finally, the dynamics of a slightly perturbed
standing wave is numerically investigated by means a highly acccurate Fourier
solver.Comment: 33 pages, 10 figures, 70 references. Other author's papers can be
found at http://www.denys-dutykh.com
Quantization of Field Theories Generalizing Gravity-Yang-Mills Systems on the Cylinder
Pure gravity and gauge theories in two dimensions are shown to be special
cases of a much more general class of field theories each of which is
characterized by a Poisson structure on a finite dimensional target space. A
general scheme for the quantization of these theories is formulated. Explicit
examples are studied in some detail. In particular gravity and gauge theories
with equivalent actions are compared. Big gauge transformations as well as the
condition of metric nondegeneracy in gravity turn out to cause significant
differences in the structure of the corresponding reduced phase spaces and the
quantum spectra of Dirac observables. For gravity coupled to SU(2) Yang
Mills the question of quantum dynamics (`problem of time') is addressed. [This
article is a contribution to the proceedings (to appear in LNP) of the 3rd
Baltic RIM Student Seminar (1993). Importance is attached to concrete examples.
A more abstract presentation of the ideas underlying this article (including
new developments) is found in hep-th/9405110.]Comment: 26, pages, TUW-94-
Separate variable blow-up patterns for a reaction-diffusion equation with critical weighted reaction
We study the separate variable blow-up patterns associated to the following
second order reaction-diffusion equation: posed for , , where ,
dimension and . A new and explicit critical exponent is introduced and a classification of the
blow-up profiles is given. The most interesting contribution of the paper is
showing that existence and behavior of the blow-up patterns is split into
different regimes by the critical exponent and also depends strongly
on whether the dimension or . These results extend
previous works of the authors in dimension
The Dirac point electron in zero-gravity Kerr--Newman spacetime
Dirac's wave equation for a point electron in the topologically nontrivial
maximal analytically extended electromagnetic Kerr--Newman spacetime is studied
in a zero-gravity limit; here, "zero-gravity" means , where is
Newton's constant of universal gravitation. The following results are obtained:
the formal Dirac Hamiltonian on the static spacelike slices is essentially
self-adjoint; the spectrum of the self-adjoint extension is symmetric about
zero, featuring a continuum with a gap about zero that, under two smallness
conditions, contains a point spectrum. Some of our results extend to a
generalization of the zero- Kerr--Newman spacetime with different
electric-monopole-to-magnetic-dipole-moment ratio.Comment: 49 pages, 17 figures; referee's comments implemented; the endnotes in
the published version appear as footnotes in this preprin
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