Block-Diagonalization of the Symmetric First-Order Coupled-Mode System

Chugunova, Marina; Pelinovsky, Dmitry E.

doi:10.1137/050629781

Cited by 36 publications

(72 citation statements)

References 24 publications

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“…This behavior is similar to the stability analysis of the coupled-mode system (9) with γ 0 = 0 and γ 1 = 0 (see [31] and references therein).…”

Section: Stabilitysupporting

confidence: 56%

“…(10) satisfies the assumption on symmetric potential functions used recently for analyzing the existence and stability of gap solitons [31]. While the previous work concerned gap solitons in coupled-mode equations with cubic nonlinearity, we will focus on the new effects that arise from the quintic nonlinear terms.…”

Section: Coupled-mode Equationsmentioning

confidence: 99%

“…This prevents the appearance of spurious complex eigenvalues that may otherwise arise from the discretization of the continuous spectrum. Moreover, by using the block-diagonalization in (26)- (27), we are able to reduce the memory constraints and double the speed of the numerical computations (see the details in [31]). The numerical eigenvalues of the operators L, H + , and H − are displayed in Fig.…”

Section: Stabilitymentioning

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].…”

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confidence: 99%

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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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“…This behavior is similar to the stability analysis of the coupled-mode system (9) with γ 0 = 0 and γ 1 = 0 (see [31] and references therein).…”

Section: Stabilitysupporting

confidence: 56%

Section: Coupled-mode Equationsmentioning

confidence: 99%

Section: Stabilitymentioning

confidence: 99%

mentioning

confidence: 99%

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Feshbach resonance management of Bose-Einstein condensates in optical lattices

2006

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“…Numerical results [BC12a] show that in the 1D Soler model (cubic nonlinearity) all solitary waves are spectrally stable. We also mention the related numerical results in [CP06,Chu08].…”

Section: Introductionmentioning

confidence: 99%

On linear instability of solitary waves for the nonlinear Dirac equation

Comech

Guan

Gustafson

2014

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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Abstract. We consider the nonlinear Dirac equation, also known as the Soler model:We study the point spectrum of linearizations at solitary waves that bifurcate from NLS solitary waves in the limit ω → m, proving that if k > 2/n, then one positive and one negative eigenvalue are present in the spectrum of the linearizations at these solitary waves with ω sufficiently close to m, so that these solitary waves are linearly unstable. The approach is based on applying the Rayleigh-Schrödinger perturbation theory to the nonrelativistic limit of the equation. The results are in formal agreement with the Vakhitov-Kolokolov stability criterion.Résumé. Nous considérons l'équation de Dirac non linéaire, aussi connu comme le modèle de Soler. Nous étudions le spectre ponctuel des linéarisations aux ondes solitaires des petites amplitudes dans la limite ω → m, et montrons que si k > 2/n, ensuite une valeur propre positive et une négative sont présents dans le spectre des linéarisations à ces ondes solitaires lorsque ω est suffisamment proche de m, ensuite ces ondes solitaires sont linéairement instable. L'approche est basée sur l'application de théorie de la perturbation de Rayleigh-Schrödinger à la limite non relativiste de l'équation. Les résultats sont en accord formel avec le critère de stabilité de Vakhitov-Kolokolov.

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Moving gap solitons in periodic potentials

Pelinovsky

Schneider

2008

Math Methods in App Sciences

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We address existence of moving gap solitons (traveling localized solutions) in the Gross-Pitaevskii equation with a small periodic potential. Moving gap solitons are approximated by the explicit localized solutions of the coupled-mode system. We show however that exponentially decaying traveling solutions of the Gross-Pitaevskii equation do not generally exist in the presence of a periodic potential due to bounded oscillatory tails ahead and behind the moving solitary waves. The oscillatory tails are not accounted in the coupled-mode formalism and are estimated by using techniques of spatial dynamics and local center-stable manifold reductions. Existence of bounded traveling solutions of the Gross--Pitaevskii equation with a single bump surrounded by oscillatory tails on a finite large interval of the spatial scale is proven by using these technique. We also show generality of oscillatory tails in other nonlinear equations with a periodic potential.Comment: 22 pages, 2 figure

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Block-Diagonalization of the Symmetric First-Order Coupled-Mode System

Cited by 36 publications

References 24 publications

Feshbach resonance management of Bose-Einstein condensates in optical lattices

Feshbach resonance management of Bose-Einstein condensates in optical lattices

On linear instability of solitary waves for the nonlinear Dirac equation

Moving gap solitons in periodic potentials

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