143 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
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
Separatrix splitting at a Hamiltonian bifurcation
We discuss the splitting of a separatrix in a generic unfolding of a
degenerate equilibrium in a Hamiltonian system with two degrees of freedom. We
assume that the unperturbed fixed point has two purely imaginary eigenvalues
and a double zero one. It is well known that an one-parametric unfolding of the
corresponding Hamiltonian can be described by an integrable normal form. The
normal form has a normally elliptic invariant manifold of dimension two. On
this manifold, the truncated normal form has a separatrix loop. This loop
shrinks to a point when the unfolding parameter vanishes. Unlike the normal
form, in the original system the stable and unstable trajectories of the
equilibrium do not coincide in general. The splitting of this loop is
exponentially small compared to the small parameter. This phenomenon implies
non-existence of single-round homoclinic orbits and divergence of series in the
normal form theory. We derive an asymptotic expression for the separatrix
splitting. We also discuss relations with behaviour of analytic continuation of
the system in a complex neighbourhood of the equilibrium
Vortices in Bose-Einstein Condensates: Some Recent Developments
In this brief review we summarize a number of recent developments in the
study of vortices in Bose-Einstein condensates, a topic of considerable
theoretical and experimental interest in the past few years. We examine the
generation of vortices by means of phase imprinting, as well as via dynamical
instabilities. Their stability is subsequently examined in the presence of
purely magnetic trapping, and in the combined presence of magnetic and optical
trapping. We then study pairs of vortices and their interactions, illustrating
a reduced description in terms of ordinary differential equations for the
vortex centers. In the realm of two vortices we also consider the existence of
stable dipole clusters for two-component condensates. Last but not least, we
discuss mesoscopic patterns formed by vortices, the so-called vortex lattices
and analyze some of their intriguing dynamical features. A number of
interesting future directions are highlighted.Comment: 24 pages, 8 figs, ws-mplb.cls, to appear in Modern Physics Letters B
(2005
Solitary Waves Under the Competition of Linear and Nonlinear Periodic Potentials
In this paper, we study the competition of linear and nonlinear lattices and
its effects on the stability and dynamics of bright solitary waves. We consider
both lattices in a perturbative framework, whereby the technique of Hamiltonian
perturbation theory can be used to obtain information about the existence of
solutions, and the same approach, as well as eigenvalue count considerations,
can be used to obtained detailed conditions about their linear stability. We
find that the analytical results are in very good agreement with our numerical
findings and can also be used to predict features of the dynamical evolution of
such solutions.Comment: 13 pages, 4 figure
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