Bifurcations and stability of gap solitons in periodic potentials

Pelinovsky, Dmitry E.; Sukhorukov, Andrey A.; Kivshar, Yuri S.

doi:10.1103/physreve.70.036618

Cited by 163 publications

(208 citation statements)

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“…Generically, the equivalent spectral condition (8) is satisfied when the soliton is centered at a local minimum of the potential, but violated when the soliton is centered at a local maximum or saddle point of the potential [33,34,35,36,37,49,58,59,60]. Although generically λ 1.…”

Section: B Stability Conditions In Inhomogeneous Mediamentioning

confidence: 99%

Qualitative and quantitative analysis of stability and instability dynamics of positive lattice solitons

Sivan

Fibich²,

Ilan³

et al. 2008

Phys. Rev. E

103

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We present a unified approach for qualitative and quantitative analysis of stability and instability dynamics of positive bright solitons in multi-dimensional focusing nonlinear media with a potential (lattice), which can be periodic, periodic with defects, quasiperiodic, single waveguide, etc. We show that when the soliton is unstable, the type of instability dynamic that develops depends on which of two stability conditions is violated. Specifically, violation of the slope condition leads to a focusing instability, whereas violation of the spectral condition leads to a drift instability. We also present a quantitative approach that allows to predict the stability and instability strength.

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Section: B Stability Conditions In Inhomogeneous Mediamentioning

confidence: 99%

Qualitative and quantitative analysis of stability and instability dynamics of positive lattice solitons

Sivan

Fibich²,

Ilan³

et al. 2008

Phys. Rev. E

103

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“…Therefore, the gap soliton solutions bifurcate from the lower spectral band via a local (small-amplitude) bifurcation. They terminate at the upper spectral band via a nonlocal (largeamplitude) bifurcation, similar to gap solitons in the GP equation with a periodic potential [27]. …”

mentioning

confidence: 99%

“…We then showed with numerical simulations of the averaged Gross-Pitaevsky (GP) equation that unstable gap solitons exhibit beating between different localized shapes, thereby confirming the stability results predicted from the coupled-mode theory. We corroborated this further with numerical simulations of the original GP equation, which show that the stable gap solitons persist much longer than the unstable ones.We note that gap solitons in the GP equation with a periodic potential (6) and the coupled-mode system (9) in the case of no Feshbach resonance management have been studied recently in [27] and [31], respectively. We can see from comparing the previous and new results that Feshbach resonance management leads to a new effect with respect to the existence of the power threshold near the local bifurcation limit [33].…”

mentioning

confidence: 99%

“…(2). The nonlinear coefficient can be written [20]and the potential for optical lattices is [27] V (x) = 2ǫV 0 cos(ωx) .Here, γ 0 is the mean value of the nonlinearity coefficient; γ(τ ), with τ = t/ε, is a mean-zero periodic function with unit period; ε ≪ 1 is a small parameter describing the strength of the Feshbach resonance management; ω is the wavenumber of the optical lattice; and ǫV 0 is a small parameter describing the strength (amplitude) of the lattice. Because (3) changes rapidly in time (i.e., the management is strong), it is reasonable use a model without dissipation [24].…”

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Feshbach resonance management of Bose-Einstein condensates in optical lattices

2006

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We analyze gap solitons in trapped Bose-Einstein condensates (BECs) in optical lattice potentials under Feshbach resonance management. Starting with an averaged Gross-Pitaevsky (GP) equation with a periodic potential, we employ an envelope wave approximation to derive coupled-mode equations describing the slow BEC dynamics in the first spectral gap of the optical lattice. We construct exact analytical formulas describing gap soliton solutions and examine their spectral stability using the Chebyshev interpolation method. We show that these gap solitons are unstable far from the threshold of local bifurcation and that the instability results in the distortion of their shape. We also predict the threshold of the power of gap solitons near the local bifurcation limit. INTRODUCTIONAt sufficiently low temperatures, particles in a dilute boson gas can condense in the ground state, forming a Bose-Einstein condensate (BEC) [1]. Under the typical confining conditions of experimental settings, BECs are inhomogeneous and the number of condensed atoms (N ) ranges from several thousand (or less) to several million (or more). The magnetic traps that confine them are usually approximated by harmonic potentials. There are two characteristic length scales: the harmonic oscillator length a ho = /(mω ho ) [which is on the order of a few microns], where ω ho = (ω x ω y ω z ) 1/3 is the geometric mean of the trapping frequencies, and the mean healing length χ = 1/ 8π|a|n [which is on the order of a micron], wheren is the mean particle density and a, the (two-body) s-wave scattering length, is determined by the atomic species of the condensate. Interactions between atoms are repulsive when a > 0 and attractive when a < 0. For a dilute ideal gas, a ≈ 0.If considering only two-body, mean-field interactions, a dilute Bose-Einstein gas can be modeled using a cubic nonlinear Schrödinger equation (NLS) with an external potential; this is also known as the Gross-Pitaevsky (GP) equation. BECs are modeled in the quasi-onedimensional (quasi-1D) regime when the transverse dimensions of the condensate are on the order of its healing length and its longitudinal dimension is much larger than its transverse ones [2]. The GP equation for the condensate wavefunction ψ(x, t) takes the formwhere |ψ| 2 is the number density, V (x) is the external trapping potential,] is proportional to the two-body scattering length, and ζ = |ψ| 2 |a| 3 is the dilute gas parameter [1,2].Experimentally realizable potentials V (x) include harmonic traps, quartic double-well traps, optical lattices and superlattices, and superpositions of lattices or superlattices with harmonic traps. The existence of quasi-1D ("cigar-shaped") BECs motivates the study of lower dimensional models such as Eq. (1). We focus here on the case of spatially periodic potentials without a confining trap along the dimension of the lattice, as that is of particular theoretical and experimental interest. For example, such potentials have been used to study Josephson effects [3], squeezed states [4], L...

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“…3(a), the dispersion is negative at the bottom of the gap and positive at the top. As the condition g 1D D < 0 is required for the formation of bright gap solitons, bright gap soliton families originate near the bottom of the gaps, for repulsive condensates (g 1D > 0), and near the top of the gaps, for attractive condensates (g 1D < 0) [21,22,23,24,25], and exist for the entire range of chemical potentials within that gap [22,23,24]. The group velocity v g = ∂µ/∂k vanishes at the band edges, hence the gap solitons form as immobile localized wavepackets.…”

Section: Matter-wave Spectrum In a Superlatticementioning

confidence: 99%

Dispersion control for matter waves and gap solitons in optical superlattices

2005

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We present a numerical study of dispersion manipulation and formation of matter-wave gap solitons in a Bose-Einstein condensate trapped in an optical superlattice. We demonstrate a method for controlled generation of matter-wave gap solitons in a stationary lattice by using an interference pattern of two condensate wavepackets, which mimics the structure of the gap soliton near the edge of a spectral band. The efficiency of this method is compared with that of gap soliton generation in a moving lattice recently demonstrated experimentally by Eiermann et al. [Phys. Rev. Lett., 92, 230401 (2004)]. We show that, by changing the relative depths of the superlattice wells, one can fine-tune the effective dispersion of the matter waves at the edges of the mini-gaps of the superlattice Bloch-wave spectrum and therefore effectively control both the peak density and the spatial width of the emerging gap solitons.

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Bifurcations and stability of gap solitons in periodic potentials

Cited by 163 publications

References 51 publications

Qualitative and quantitative analysis of stability and instability dynamics of positive lattice solitons

Qualitative and quantitative analysis of stability and instability dynamics of positive lattice solitons

Feshbach resonance management of Bose-Einstein condensates in optical lattices

Dispersion control for matter waves and gap solitons in optical superlattices

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