14 research outputs found
On classification of intrinsic localized modes for the Discrete Nonlinear Schr\"{o}dinger Equation
We consider localized modes (discrete breathers) of the discrete nonlinear
Schr\"{o}dinger equation
,
, . We study the diversity of the steady-state
solutions of the form and the intervals of the
frequency, , of their existence. The base for the analysis is provided
by the anticontinuous limit ( negative and large enough) where all the
solutions can be coded by the sequences of three symbols "-", "0" and "+".
Using dynamical systems approach we show that this coding is valid for
and the point is a point of
accumulation of saddle-node bifurcations. Also we study other bifurcations of
intrinsic localized modes which take place for and give the
complete table of them for the solutions with codes consisting of less than
four symbols.Comment: 33 pages, 14 figures. To appear in Physica
Coupled-mode equations and gap solitons in a two-dimensional nonlinear elliptic problem with a separable periodic potential
We address a two-dimensional nonlinear elliptic problem with a
finite-amplitude periodic potential. For a class of separable symmetric
potentials, we study the bifurcation of the first band gap in the spectrum of
the linear Schr\"{o}dinger operator and the relevant coupled-mode equations to
describe this bifurcation. The coupled-mode equations are derived by the
rigorous analysis based on the Fourier--Bloch decomposition and the Implicit
Function Theorem in the space of bounded continuous functions vanishing at
infinity. Persistence of reversible localized solutions, called gap solitons,
beyond the coupled-mode equations is proved under a non-degeneracy assumption
on the kernel of the linearization operator. Various branches of reversible
localized solutions are classified numerically in the framework of the
coupled-mode equations and convergence of the approximation error is verified.
Error estimates on the time-dependent solutions of the Gross--Pitaevskii
equation and the coupled-mode equations are obtained for a finite-time
interval.Comment: 32 pages, 16 figure
One-dimension cubic-quintic Gross-Pitaevskii equation in Bose-Einstein condensates in a trap potential
By means of new general variational method we report a direct solution for
the quintic self-focusing nonlinearity and cubic-quintic 1D Gross Pitaeskii
equation (GPE) in a harmonic confined potential. We explore the influence of
the 3D transversal motion generating a quintic nonlinear term on the ideal 1D
pure cigar-like shape model for the attractive and repulsive atom-atom
interaction in Bose Einstein condensates (BEC). Also, we offer a closed
analytical expression for the evaluation of the error produced when solely the
cubic nonlinear GPE is considered for the description of 1D BEC.Comment: 6 pages, 3 figure
Solitary waves for linearly coupled nonlinear Schrodinger equations with inhomogeneous coefficients
Motivated by the study of matter waves in Bose-Einstein condensates and
coupled nonlinear optical systems, we study a system of two coupled nonlinear
Schrodinger equations with inhomogeneous parameters, including a linear
coupling. For that system we prove the existence of two different kinds of
homoclinic solutions to the origin describing solitary waves of physical
relevance. We use a Krasnoselskii fixed point theorem together with a suitable
compactness criterion.Comment: 16 page
Resonant scattering of matter-wave gap solitons by optical lattice defects
The physical mechanism underlying scattering properties of matter wave
gap-solitons by linear optical lattice defects is investigated. The occurrence
of repeated reflection, transmission and trapping regions for increasing
strengths of an optical lattice defect are shown to be due to impurity modes
inside the defect potential with chemical potentials and numbers of atoms
matching corresponding quantities of an incoming gap-soliton. For gap-solitons
with chemical potentials very close to band edges, the number of resonances
observed in the scattering coincides with the number of bound states which can
exist in the defect potential for the given defect strength. The dependence of
the positions and widths of the transmission resonant on the incoming
gap-soliton velocities are investigated by means of a defect mode analysis and
effective mass theory. The comparisons with direct integrations of the
Gross-Pitaevskii equation provide a very good agreement confirming the
correctness of our interpretation. The possibility of multiple resonant
transmission through arrays of optical lattice defects is also demonstrated. In
particular, we show that it is possible to design the strength of the defects
so to balance the velocity detunings and to allow the resonant transmission
through a larger number of defects. The possibility of using these results for
very precise gap-soliton dynamical filters is suggested.Comment: 9 pages, 13 figure