169 research outputs found
Two-dimensional discrete solitons in dipolar Bose-Einstein condensates
We analyze the formation and dynamics of bright unstaggered solitons in the
disk-shaped dipolar Bose-Einstein condensate, which features the interplay of
contact (collisional) and long-range dipole-dipole (DD) interactions between
atoms. The condensate is assumed to be trapped in a strong optical-lattice
potential in the disk's plane, hence it may be approximated by a
two-dimensional (2D) discrete model, which includes the on-site nonlinearity
and cubic long-range (DD) interactions between sites of the lattice. We
consider two such models, that differ by the form of the on-site nonlinearity,
represented by the usual cubic term, or more accurate nonpolynomial one,
derived from the underlying 3D Gross-Pitaevskii equation. Similar results are
obtained for both models. The analysis is focused on effects of the DD
interaction on fundamental localized modes in the lattice (2D discrete
solitons). The repulsive isotropic DD nonlinearity extends the existence and
stability regions of the fundamental solitons. New families of on-site,
inter-site and hybrid solitons, built on top of a finite background, are found
as a result of the interplay of the isotropic repulsive DD interaction and
attractive contact nonlinearity. By themselves, these solutions are unstable,
but they evolve into robust breathers which exist on an oscillating background.
In the presence of the repulsive contact interactions, fundamental localized
modes exist if the DD interaction (attractive isotropic or anisotropic) is
strong enough. They are stable in narrow regions close to the anticontinuum
limit, while unstable solitons evolve into breathers. In the latter case, the
presence of the background is immaterial
Deviation from one-dimensionality in stationary properties and collisional dynamics of matter-wave solitons
By means of analytical and numerical methods, we study how the residual
three-dimensionality affects dynamics of solitons in an attractive
Bose-Einstein condensate loaded into a cigar-shaped trap. Based on an effective
1D Gross-Pitaevskii equation that includes an additional quintic self-focusing
term, generated by the tight transverse confinement, we find a family of exact
one-soliton solutions and demonstrate stability of the entire family, despite
the possibility of collapse in the 1D equation with the quintic self-focusing
nonlinearity. Simulating collisions between two solitons in the same setting,
we find a critical velocity, , below which merger of identical in-phase
solitons is observed. Dependence of on the strength of the transverse
confinement and number of atoms in the solitons is predicted by means of the
perturbation theory and investigated in direct simulations. Symmetry breaking
in collisions of identical solitons with a nonzero phase difference is also
shown in simulations and qualitatively explained by means of an analytical
approximation.Comment: 10 pages, 7 figure
Symbiotic gap and semi-gap solitons in Bose-Einstein condensates
Using the variational approximation and numerical simulations, we study
one-dimensional gap solitons in a binary Bose-Einstein condensate trapped in an
optical-lattice potential. We consider the case of inter-species repulsion,
while the intra-species interaction may be either repulsive or attractive.
Several types of gap solitons are found: symmetric or asymmetric; unsplit or
split, if centers of the components coincide or separate; intra-gap (with both
chemical potentials falling into a single bandgap) or inter-gap, otherwise. In
the case of the intra-species attraction, a smooth transition takes place
between solitons in the semi-infinite gap, the ones in the first finite
bandgap, and semi-gap solitons (with one component in a bandgap and the other
in the semi-infinite gap).Comment: 5 pages, 9 figure
Matter-wave 2D solitons in crossed linear and nonlinear optical lattices
It is demonstrated the existence of multidimensional matter-wave solitons in
a crossed optical lattice (OL) with linear OL in the direction and
nonlinear OL (NOL) in the direction, where the NOL can be generated by a
periodic spatial modulation of the scattering length using an optically induced
Feshbach resonance. In particular, we show that such crossed linear and
nonlinear OL allows to stabilize two-dimensional (2D) solitons against decay or
collapse for both attractive and repulsive interactions. The solutions for the
soliton stability are investigated analytically, by using a multi-Gaussian
variational approach (VA), with the Vakhitov-Kolokolov (VK) necessary criterion
for stability; and numerically, by using the relaxation method and direct
numerical time integrations of the Gross-Pitaevskii equation (GPE). Very good
agreement of the results corresponding to both treatments is observed.Comment: 8 pages (two-column format), with 16 eps-files of 4 figure
Universal fractal structures in the weak interaction of solitary waves in generalized nonlinear Schr\"{o}dinger equations
Weak interactions of solitary waves in the generalized nonlinear
Schr\"{o}dinger equations are studied. It is first shown that these
interactions exhibit similar fractal dependence on initial conditions for
different nonlinearities. Then by using the Karpman-Solov'ev method, a
universal system of dynamical equations is derived for the velocities,
amplitudes, positions and phases of interacting solitary waves. These dynamical
equations contain a single parameter, which accounts for the different forms of
nonlinearity. When this parameter is zero, these dynamical equations are
integrable, and the exact analytical solutions are derived. When this parameter
is non-zero, the dynamical equations exhibit fractal structures which match
those in the original wave equations both qualitatively and quantitatively.
Thus the universal nature of fractal structures in the weak interaction of
solitary waves is analytically established. The origin of these fractal
structures is also explored. It is shown that these structures bifurcate from
the initial conditions where the solutions of the integrable dynamical
equations develop finite-time singularities. Based on this observation, an
analytical criterion for the existence and locations of fractal structures is
obtained. Lastly, these analytical results are applied to the generalized
nonlinear Schr\"{o}dinger equations with various nonlinearities such as the
saturable nonlinearity, and predictions on their weak interactions of solitary
waves are made.Comment: 22pages, 15 figure
Matter-wave vortices in cigar-shaped and toroidal waveguides
We study vortical states in a Bose-Einstein condensate (BEC) filling a
cigar-shaped trap. An effective one-dimensional (1D) nonpolynomial Schroedinger
equation (NPSE) is derived in this setting, for the models with both repulsive
and attractive inter-atomic interactions. Analytical formulas for the density
profiles are obtained from the NPSE in the case of self-repulsion within the
Thomas-Fermi approximation, and in the case of the self-attraction as exact
solutions (bright solitons). A crucially important ingredient of the analysis
is the comparison of these predictions with direct numerical solutions for the
vortex states in the underlying 3D Gross-Pitaevskii equation (GPE). The
comparison demonstrates that the NPSE provides for a very accurate
approximation, in all the cases, including the prediction of the stability of
the bright solitons and collapse threshold for them. In addition to the
straight cigar-shaped trap, we also consider a torus-shaped configuration. In
that case, we find a threshold for the transition from the axially uniform
state, with the transverse intrinsic vorticity, to a symmetry-breaking pattern,
due to the instability in the self-attractive BEC filling the circular trap.Comment: 6 pages, Physical Review A, in pres
Spatiotemporal discrete multicolor solitons
We have found various families of two-dimensional spatiotemporal solitons in
quadratically nonlinear waveguide arrays. The families of unstaggered odd, even
and twisted stationary solutions are thoroughly characterized and their
stability against perturbations is investigated. We show that the twisted and
even solutions display instability, while most of the odd solitons show
remarkable stability upon evolution.Comment: 18 pages,7 figures. To appear in Physical Review
Vector solitons in nearly-one-dimensional Bose-Einstein condensates
We derive a system of nonpolynomial Schroedinger equations (NPSEs) for
one-dimensional wave functions of two components in a binary self-attractive
Bose-Einstein condensate loaded in a cigar-shaped trap. The system is obtained
by means of the variational approximation, starting from the coupled 3D
Gross-Pitaevskii equations and assuming, as usual, the factorization of 3D wave
functions. The system can be obtained in a tractable form under a natural
condition of symmetry between the two species. A family of vector
(two-component) soliton solutions is constructed. Collisions between orthogonal
solitons (ones belonging to the different components) are investigated by means
of simulations. The collisions are essentially inelastic. They result in strong
excitation of intrinsic vibrations in the solitons, and create a small
orthogonal component ("shadow") in each colliding soliton. The collision may
initiate collapse, which depends on the mass and velocities of the solitons.Comment: 7 pages, 6 figures; Physical Review A, in pres
Discrete Solitons and Vortices on Anisotropic Lattices
We consider effects of anisotropy on solitons of various types in
two-dimensional nonlinear lattices, using the discrete nonlinear
Schr{\"{o}}dinger equation as a paradigm model. For fundamental solitons, we
develop a variational approximation, which predicts that broad quasi-continuum
solitons are unstable, while their strongly anisotropic counterparts are
stable. By means of numerical methods, it is found that, in the general case,
the fundamental solitons and simplest on-site-centered vortex solitons ("vortex
crosses") feature enhanced or reduced stability areas, depending on the
strength of the anisotropy. More surprising is the effect of anisotropy on the
so-called "super-symmetric" intersite-centered vortices ("vortex squares"),
with the topological charge equal to the square's size : we predict in
an analytical form by means of the Lyapunov-Schmidt theory, and confirm by
numerical results, that arbitrarily weak anisotropy results in dramatic changes
in the stability and dynamics in comparison with the \emph{degenerate}, in this
case, isotropic limit.Comment: 10 pages + 7 figure
On spectral stability of solitary waves of nonlinear Dirac equation on a line
We study the spectral stability of solitary wave solutions to the nonlinear
Dirac equation in one dimension. We focus on the Dirac equation with cubic
nonlinearity, known as the Soler model in (1+1) dimensions and also as the
massive Gross-Neveu model. Presented numerical computations of the spectrum of
linearization at a solitary wave show that the solitary waves are spectrally
stable. We corroborate our results by finding explicit expressions for several
of the eigenfunctions. Some of the analytic results hold for the nonlinear
Dirac equation with generic nonlinearity.Comment: 20 pages with figure
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