164 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
Dissipative surface solitons in periodic structures
We report dissipative surface solitons forming at the interface between a
semi-infinite lattice and a homogeneous Kerr medium. The solitons exist due to
balance between amplification in the near-surface lattice channel and
two-photon absorption. The stable dissipative surface solitons exist in both
focusing and defocusing media, when propagation constants of corresponding
states fall into a total semi-infinite and or into one of total finite gaps of
the spectrum (i.e. in a domain where propagation of linear waves is inhibited
for the both media). In a general situation, the surface solitons form when
amplification coefficient exceeds threshold value. When a soliton is formed in
a total finite gap there exists also the upper limit for the linear gain.Comment: 5 pages, 3 figures, to appear in Europhysics Letter
Wannier functions analysis of the nonlinear Schr\"{o}dinger equation with a periodic potential
In the present Letter we use the Wannier function basis to construct lattice
approximations of the nonlinear Schr\"{o}dinger equation with a periodic
potential. We show that the nonlinear Schr\"{o}dinger equation with a periodic
potential is equivalent to a vector lattice with long-range interactions. For
the case-example of the cosine potential we study the validity of the so-called
tight-binding approximation i.e., the approximation when nearest neighbor
interactions are dominant. The results are relevant to Bose-Einstein condensate
theory as well as to other physical systems like, for example, electromagnetic
wave propagation in nonlinear photonic crystals.Comment: 5 pages, 1 figure, submitted to Phys. Rev.
Mixed symmetry localized modes and breathers in binary mixtures of Bose-Einstein condensates in optical lattices
We study localized modes in binary mixtures of Bose-Einstein condensates
embedded in one-dimensional optical lattices. We report a diversity of
asymmetric modes and investigate their dynamics. We concentrate on the cases
where one of the components is dominant, i.e. has much larger number of atoms
than the other one, and where both components have the numbers of atoms of the
same order but different symmetries. In the first case we propose a method of
systematic obtaining the modes, considering the "small" component as
bifurcating from the continuum spectrum. A generalization of this approach
combined with the use of the symmetry of the coupled Gross-Pitaevskii equations
allows obtaining breather modes, which are also presented.Comment: 11 pages, 16 figure
Quantum signatures of breather-breather interactions
The spectrum of the Quantum Discrete Nonlinear Schr\"odinger equation on a
periodic 1D lattice shows some interesting detailed band structure which may be
interpreted as the quantum signature of a two-breather interaction in the
classical case. We show that this fine structure can be interpreted using
degenerate perturbation theory.Comment: 4 pages, 4 fig
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