189 research outputs found
Nonlinear modes for the Gross-Pitaevskii equation -- demonstrative computation approach
A method for the study of steady-state nonlinear modes for Gross-Pitaevskii
equation (GPE) is described. It is based on exact statement about coding of the
steady-state solutions of GPE which vanish as by reals. This
allows to fulfill {\it demonstrative computation} of nonlinear modes of GPE
i.e. the computation which allows to guarantee that {\it all} nonlinear modes
within a given range of parameters have been found. The method has been applied
to GPE with quadratic and double-well potential, for both, repulsive and
attractive nonlinearities. The bifurcation diagrams of nonlinear modes in these
cases are represented. The stability of these modes has been discussed.Comment: 21 pages, 6 figure
Coupled oscillators with power-law interaction and their fractional dynamics analogues
The one-dimensional chain of coupled oscillators with long-range power-law
interaction is considered. The equation of motion in the infrared limit are
mapped onto the continuum equation with the Riesz fractional derivative of
order , when . The evolution of soliton-like and
breather-like structures are obtained numerically and compared for both types
of simulations: using the chain of oscillators and using the continuous medium
equation with the fractional derivative.Comment: 16 pages, 5 figure
Pattern Forming Dynamical Instabilities of Bose-Einstein Condensates: A Short Review
In this short topical review, we revisit a number of works on the
pattern-forming dynamical instabilities of Bose-Einstein condensates in one-
and two-dimensional settings. In particular, we illustrate the trapping
conditions that allow the reduction of the three-dimensional, mean field
description of the condensates (through the Gross-Pitaevskii equation) to such
lower dimensional settings, as well as to lattice settings. We then go on to
study the modulational instability in one dimension and the snaking/transverse
instability in two dimensions as typical examples of long-wavelength
perturbations that can destabilize the condensates and lead to the formation of
patterns of coherent structures in them. Trains of solitons in one-dimension
and vortex arrays in two-dimensions are prototypical examples of the resulting
nonlinear waveforms, upon which we briefly touch at the end of this review.Comment: 28 pages, 9 figures, publishe
Regular spatial structures in arrays of Bose-Einstein condensates induced by modulational instability
We show that the phenomenon of modulational instability in arrays of
Bose-Einstein condensates confined to optical lattices gives rise to coherent
spatial structures of localized excitations. These excitations represent thin
disks in 1D, narrow tubes in 2D, and small hollows in 3D arrays, filled in with
condensed atoms of much greater density compared to surrounding array sites.
Aspects of the developed pattern depend on the initial distribution function of
the condensate over the optical lattice, corresponding to particular points of
the Brillouin zone. The long-time behavior of the spatial structures emerging
due to modulational instability is characterized by the periodic recurrence to
the initial low-density state in a finite optical lattice. We propose a simple
way to retain the localized spatial structures with high atomic concentration,
which may be of interest for applications. Theoretical model, based on the
multiple scale expansion, describes the basic features of the phenomenon.
Results of numerical simulations confirm the analytical predictions.Comment: 17 pages, 13 figure
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