20,248 research outputs found
Stability analysis of the D-dimensional nonlinear Schroedinger equation with trap and two- and three-body interactions
Considering the static solutions of the D-dimensional nonlinear Schroedinger
equation with trap and attractive two-body interactions, the existence of
stable solutions is limited to a maximum critical number of particles, when D
is greater or equal 2. In case D=2, we compare the variational approach with
the exact numerical calculations. We show that, the addition of a positive
three-body interaction allows stable solutions beyond the critical number. In
this case, we also introduce a dynamical analysis of the conditions for the
collapse.Comment: 6 pages, 7 figure
Relation between the eigenfrequencies of Bogoliubov excitations of Bose-Einstein condensates and the eigenvalues of the Jacobian in a time-dependent variational approach
We study the relation between the eigenfrequencies of the Bogoliubov
excitations of Bose-Einstein condensates, and the eigenvalues of the Jacobian
stability matrix in a variational approach which maps the Gross-Pitaevskii
equation to a system of equations of motion for the variational parameters. We
do this for Bose-Einstein condensates with attractive contact interaction in an
external trap, and for a simple model of a self-trapped Bose-Einstein
condensate with attractive 1/r interaction. The stationary solutions of the
Gross-Pitaevskii equation and Bogoliubov excitations are calculated using a
finite-difference scheme. The Bogoliubov spectra of the ground and excited
state of the self-trapped monopolar condensate exhibits a Rydberg-like
structure, which can be explained by means of a quantum defect theory. On the
variational side, we treat the problem using an ansatz of time-dependent
coupled Gaussians combined with spherical harmonics. We first apply this ansatz
to a condensate in an external trap without long-range interaction, and
calculate the excitation spectrum with the help of the time-dependent
variational principle. Comparing with the full-numerical results, we find a
good agreement for the eigenfrequencies of the lowest excitation modes with
arbitrary angular momenta. The variational method is then applied to calculate
the excitations of the self-trapped monopolar condensates, and the
eigenfrequencies of the excitation modes are compared.Comment: 15 pages, 12 figure
Reduced dimensionality and spatial entanglement in highly anisotropic Bose-Einstein condensates
We investigate the reduced dimensionality of highly anisotropic Bose-Einstein
condensates (BECs) in connection to the entanglement between its spatial
degrees of freedom. We argue that the reduced-dimensionality of the BEC is
physically meaningful in a regime where spatial correlations are negligible. We
handle the problem analytically within the mean-field approximation for general
quasi-one-dimensional and -two-dimensional geometries, and obtain the optimal
reduced-dimension, pure-state description of the condensate mean field. We give
explicit solutions to the case of harmonic potentials, which we compare against
exact numerical integration of the three-dimensional Gross-Pitaevskii equation.Comment: 15 pages, 3 figures. Minor changes in text to be in agreement with
published versio
A basis for variational calculations in d dimensions
In this paper we derive expressions for matrix elements (\phi_i,H\phi_j) for
the Hamiltonian H=-\Delta+\sum_q a(q)r^q in d > 1 dimensions.
The basis functions in each angular momentum subspace are of the form
phi_i(r)=r^{i+1+(t-d)/2}e^{-r^p/2}, i >= 0, p > 0, t > 0. The matrix elements
are given in terms of the Gamma function for all d. The significance of the
parameters t and p and scale s are discussed. Applications to a variety of
potentials are presented, including potentials with singular repulsive terms of
the form b/r^a, a,b > 0, perturbed Coulomb potentials -D/r + B r + Ar^2, and
potentials with weak repulsive terms, such as -g r^2 + r^4, g > 0.Comment: 22 page
Exact solutions of semilinear radial wave equations in n dimensions
Exact solutions are derived for an n-dimensional radial wave equation with a
general power nonlinearity. The method, which is applicable more generally to
other nonlinear PDEs, involves an ansatz technique to solve a first-order PDE
system of group-invariant variables given by group foliations of the wave
equation, using the one-dimensional admitted point symmetry groups. (These
groups comprise scalings and time translations, admitted for any nonlinearity
power, in addition to space-time inversions admitted for a particular conformal
nonlinearity power). This is shown to yield not only group-invariant solutions
as derived by standard symmetry reduction, but also other exact solutions of a
more general form. In particular, solutions with interesting analytical
behavior connected with blow ups as well as static monopoles are obtained.Comment: 29 pages, 1 figure. Published version with minor correction
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