138 research outputs found
Modulated amplitude waves with nonzero phases in Bose-Einstein condensates
In this paper we give a frame for application of the averaging method to
Bose-Einstein condensates (BECs) and obtain an abstract result upon the
dynamics of BECs. Using aver- aging method, we determine the location where the
modulated amplitude waves (periodic or quasi-periodic) exist and we also study
the stability and instability of modulated amplitude waves (periodic or
quasi-periodic). Compared with the previous work, modulated amplitude waves
studied in this paper have nontrivial phases and this makes the problem become
more diffcult, since it involves some singularities.Comment: 17 pages, 2 figure
Loop structure of the lowest Bloch band for a Bose-Einstein condensate
We investigate analytically and numerically Bloch waves for a Bose--Einstein
condensate in a sinusoidal external potential. At low densities the dependence
of the energy on the quasimomentum is similar to that for a single particle,
but at densities greater than a critical one the lowest band becomes
triple-valued near the boundary of the first Brillouin zone and develops the
structure characteristic of the swallow-tail catastrophe. We comment on the
experimental consequences of this behavior.Comment: 4 pages, 7 figure
Controlled Generation of Dark Solitons with Phase Imprinting
The generation of dark solitons in Bose-Einstein condensates with phase
imprinting is studied by mapping it into the classic problem of a damped driven
pendulum. We provide simple but powerful schemes of designing the phase imprint
for various desired outcomes. We derive a formula for the number of dark
solitons generated by a given phase step, and also obtain results which explain
experimental observations.Comment: 4pages, 4 figure
Suppression of transverse instabilities of dark solitons and their dispersive shock waves
We investigate the impact of nonlocality, owing to diffusive behavior, on
transverse instabilities of a dark stripe propagating in a defocusing cubic
medium. The nonlocal response turns out to have a strongly stabilizing effect
both in the case of a single soliton input and in the regime where dispersive
shock waves develop "multisoliton regime". Such conclusions are supported by
the linear stability analysis and numerical simulation of the propagation
Fast soliton scattering by delta impurities
We study the Gross-Pitaevskii equation (nonlinear Schroedinger equation) with
a repulsive delta function potential. We show that a high velocity incoming
soliton is split into a transmitted component and a reflected component. The
transmitted mass (L^2 norm squared) is shown to be in good agreement with the
quantum transmission rate of the delta function potential. We further show that
the transmitted and reflected components resolve into solitons plus dispersive
radiation, and quantify the mass and phase of these solitons.Comment: 32 pages, 3 figure
Tunable tunneling: An application of stationary states of Bose-Einstein condensates in traps of finite depth
The fundamental question of how Bose-Einstein condensates tunnel into a
barrier is addressed. The cubic nonlinear Schrodinger equation with a finite
square well potential, which models a Bose-Einstein condensate in a
quasi-one-dimensional trap of finite depth, is solved for the complete set of
localized and partially localized stationary states, which the former evolve
into when the nonlinearity is increased. An immediate application of these
different solution types is tunable tunneling. Magnetically tunable Feshbach
resonances can change the scattering length of certain Bose-condensed atoms,
such as Rb, by several orders of magnitude, including the sign, and
thereby also change the mean field nonlinearity term of the equation and the
tunneling of the wavefunction. We find both linear-type localized solutions and
uniquely nonlinear partially localized solutions where the tails of the
wavefunction become nonzero at infinity when the nonlinearity increases. The
tunneling of the wavefunction into the non-classical regime and thus its
localization therefore becomes an external experimentally controllable
parameter.Comment: 11 pages, 5 figure
Uncertainty estimates and L_2 bounds for the Kuramoto-Sivashinsky equation
We consider the Kuramoto-Sivashinsky (KS) equation in one spatial dimension
with periodic boundary conditions. We apply a Lyapunov function argument
similar to the one first introduced by Nicolaenko, Scheurer, and Temam, and
later improved by Collet, Eckmann, Epstein and Stubbe, and Goodman, to prove
that ||u||_2 < C L^1.5. This result is slightly weaker than that recently
announced by Giacomelli and Otto, but applies in the presence of an additional
linear destabilizing term. We further show that for a large class of Lyapunov
functions \phi the exponent 1.5 is the best possible from this line of
argument. Further, this result together with a result of Molinet gives an
improved estimate for L_2 boundedness of the Kuramoto-Sivashinsky equation in
thin rectangular domains in two spatial dimensions.Comment: 17 pages, 1 figure; typos corrected, references added; figure
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Modulational instability in cigar shaped Bose-Einstein condensates in optical lattices
A self consistent theory of a cigar shaped Bose-Einstein condensate (BEC)
periodically modulated by a laser beam is presented. We show, both
theoretically and numerically, that modulational instability/stability is the
mechanism by which wavefunctions of soliton type can be generated in cigar
shaped BEC subject to a 1D optical lattice. The theory explains why bright
solitons can exist in BEC with positive scattering length and why condensate
with negative scattering length can be stable and give rise to dark solitary
pulses.Comment: Submitted, 4 pages, 3 figures. Revised versio
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