938 research outputs found
Soliton splitting by external delta potentials
We show that a soliton scattered by an external delta potential splits into
two solitons and a radiation term. Theoretical analysis gives the amplitudes
and phases of the reflected and transmitted solitons with errors going to zero
as the velocity of the incoming soliton tends to infinity. Numerical analysis
shows that this asymptotic relation is valid for all but very slow solitons. We
also show that the total transmitted mass, that is the square of the norm
of the solution restricted on the transmitted side of the delta potential is in
good agreement with the quantum transmission rate of the delta potential.
This paper is a numerical companion to our analytical paper on the same
topic, "Fast soliton scattering by delta impurities," math.AP/0602187
The inverse problem for the Gross - Pitaevskii equation
Two different methods are proposed for the generation of wide classes of
exact solutions to the stationary Gross - Pitaevskii equation (GPE). The first
method, suggested by the work by Kondrat'ev and Miller (1966), applies to
one-dimensional (1D) GPE. It is based on the similarity between the GPE and the
integrable Gardner equation, all solutions of the latter equation (both
stationary and nonstationary ones) generating exact solutions to the GPE, with
the potential function proportional to the corresponding solutions. The second
method is based on the "inverse problem" for the GPE, i.e. construction of a
potential function which provides a desirable solution to the equation.
Systematic results are presented for 1D and 2D cases. Both methods are
illustrated by a variety of localized solutions, including solitary vortices,
for both attractive and repulsive nonlinearity in the GPE. The stability of the
1D solutions is tested by direct simulations of the time-dependent GPE
Enhanced Quantum Reflection of Matter-Wave Solitons
Matter-wave bright solitons are predicted to reflect from a purely attractive
potential well although they are macroscopic objects with classical
particle-like properties. The non-classical reflection occurs at small
velocities and a pronounced switching to almost perfect transmission above a
critical velocity is found, caused by nonlinear mean-field interactions. Full
numerical results from the nonlinear Schr\"{o}dinger equation are complimented
by a two-mode variational calculation to explain the predicted effect, which
can be used for velocity filtering of solitons. The experimental realization
with laser-induced potentials or two-component Bose-Einstein condensates is
suggested.Comment: 7 pages, 3 figures, to be published in Europhys. Let
Low-density, one-dimensional quantum gases in a split trap
We investigate degenerate quantum gases in one dimension trapped in a
harmonic potential that is split in the centre by a pointlike potential. Since
the single particle eigenfunctions of such a system are known for all strengths
of the central potential, the dynamics for non-interacting fermionic gases and
low-density, strongly interacting bosonic gases can be investigated exactly
using the Fermi-Bose mapping theorem. We calculate the exact many-particle
ground-state wave-functions for both particle species, investigate soliton-like
solutions, and compare the bosonic system to the well-known physics of Bose
gases described by the Gross-Pitaevskii equation. We also address the
experimentally important questions of creation and detection of such states.Comment: 7 pages, 5 figure
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