An effective potential for one-dimensional matter-wave solitons in an axially inhomogeneous trap

Nicola, Sergio De; Malomed, Boris A.; Fedele, R.

doi:10.1016/j.physleta.2006.07.062

Cited by 33 publications

(42 citation statements)

References 24 publications

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“…The dynamics of Bose-Einstein condensates (BECs) made of dilute ultracold gases of bosonic atoms obeys the 3D Gross-Pitaevskii equation (GPE) in the mean-field approximation [1]. An effective 1D equation can be derived from the 3D GPE to describe the BEC dynamics in prolate ("cigar-shaped") traps [2]- [8]. In the simplest case, which corresponds to a sufficiently low BEC density, the reduced 1D equation amounts to the cubic nonlinear Schrödinger equation (NLSE) [9].…”

Section: Introductionmentioning

confidence: 99%

Two routes to the one-dimensional discrete nonpolynomial Schrödinger equation

Gligorić

Maluckov

Salasnich

et al. 2009

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The Bose-Einstein condensate (BEC), confined in a combination of the cigar-shaped trap and axial optical lattice, is studied in the framework of two models described by two versions of the onedimensional (1D) discrete nonpolynomial Schrödinger equation (NPSE). Both models are derived from the three-dimensional Gross-Pitaevskii equation (3D GPE). To produce "model 1" (which was derived in recent works), the 3D GPE is first reduced to the 1D continual NPSE, which is subsequently discretized. "Model 2", that was not considered before, is derived by first discretizing the 3D GPE, which is followed by the reduction of the dimension. The two models seem very different; in particular, model 1 is represented by a single discrete equation for the 1D wave function, while model 2 includes an additional equation for the transverse width. Nevertheless, numerical analyses show similar behaviors of fundamental unstaggered solitons in both systems, as concerns their existence region and stability limits. Both models admit the collapse of the localized modes, reproducing the fundamental property of the self-attractive BEC confined in tight traps. Thus, we conclude that the fundamental properties of discrete solitons predicted for the strongly trapped self-attracting BEC are reliable, as the two distinct models produce them in a nearly identical form. However, a difference between the models is found too, as strongly pinned (very narrow) discrete solitons, which were previously found in model 1, are not generated by model 2 -in fact, in agreement with the continual 1D NPSE, which does not have such solutions either. In that respect, the newly derived model provides for a more accurate approximation for the trapped BEC. PACS numbers: 03.75.Lm; 05.45.YvThe dynamics of a dilute quantum gas which forms the Bose-Einstein condensate (BEC) is very accurately described by the three-dimensional Gross-Pitaevskii equation (3D GPE). This equation treats effects of collisions between atoms in the condensate in the mean-field approximation. In experimentally relevant settings, the BEC is always confined by a trapping potential. In many cases, the trap is designed to have the "cigar-shaped" form, allowing an effective reduction of the dimension from 3 to 1. In turn, the 1D dynamics of the trapped condensate may be controlled by means of an additional periodic potential, induced by an optical lattice (OL), which acts along the axis of the "cigar". If the OL potential is sufficiently strong, the eventual dynamical model reduces to a 1D discrete equation. In both the continual and discrete versions of the 1D description, a crucially important feature is the form of the nonlinearity in the respective equations. In the limit of low density, the nonlinearity is cubic -the same as in the underlying 3D GPE. In the general case, a consistent derivation, which starts from the cubic nonlinearity in 3D, leads to 1D equations with a nonpolynomial nonlinearity, the respective model being called the "nonpolynomial Schrödinger equation" (NPSE). The discrete lim...

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Section: Introductionmentioning

confidence: 99%

Two routes to the one-dimensional discrete nonpolynomial Schrödinger equation

Gligorić

Maluckov

Salasnich

et al. 2009

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…The 3D GP equation (1) can be reduced to an effectively 1D form provided the transverse dimensions of the BEC are on the order of its healing length and its longitudinal dimension is much larger than the transverse ones [11,31]. In this regime, the 1D approximation is derived by averaging equation (1) in the transverse plane.…”

Section: The Perturbed Gross-pitaevskii Equationmentioning

confidence: 99%

“…Alternatively, one can use a Feshbach resonance induced by an optical [8] or dc electric [9] field. In low-dimensional settings, one can also tune the effective nonlinearity by changing the BEC's transversal confinement [10,11].…”

Section: Introductionmentioning

confidence: 99%

Modulated amplitude waves in collisionally inhomogeneous Bose–Einstein condensates

Porter

Kevrekidis

Malomed

et al. 2007

Physica D: Nonlinear Phenomena

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We investigate the dynamics of an effectively one-dimensional Bose-Einstein condensate (BEC) with scattering length a subjected to a spatially periodic modulation, a = a(x) = a(x + L). This "collisionally inhomogeneous" BEC is described by a Gross-Pitaevskii (GP) equation whose nonlinearity coefficient is a periodic function of x. We transform this equation into a GP equation with a constant coefficient and an additional effective potential and study a class of extended wave solutions of the transformed equation. For weak underlying inhomogeneity, the effective potential takes a form resembling a superlattice, and the amplitude dynamics of the solutions of the constant-coefficient GP equation obey a nonlinear generalization of the Ince equation. In the small-amplitude limit, we use averaging to construct analytical solutions for modulated amplitude waves (MAWs), whose stability we subsequently examine using both numerical simulations of the original GP equation and fixed-point computations with the MAWs as numerically exact solutions. We show that "on-site" solutions, whose maxima correspond to maxima of a(x), are more robust and likely to be observed than their "off-site" counterparts.

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“…Gap solitons are represented by stationary solutions to the respective Gross-Pitaevskii equation (GPE), with the eigenvalue (chemical potential) located in a finite bandgap of the OL-induced spectrum. Note that a periodic potential emulating the OL can also be induced by the spatially-periodic modulation of the transverse trap which confines the condensate in the transverse directions [4,5].…”

Section: Introductionmentioning

confidence: 99%

Dynamics of kicked matter-wave solitons in an optical lattice

Cetoli

Salasnich

Malomed

et al. 2009

Physica D: Nonlinear Phenomena

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We investigate effects of the application of a kick to one-dimensional matter-wave solitons in a self-attractive Bose-Einstein condensate trapped in an optical lattice. The resulting soliton's dynamics is studied within the framework of the time-dependent nonpolynomial Schrodinger equation. The crossover from the pinning to quasi-free motion crucially depends on the size of the kick, strength of the self-attraction, and parameters of the optical lattice

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An effective potential for one-dimensional matter-wave solitons in an axially inhomogeneous trap

Cited by 33 publications

References 24 publications

Two routes to the one-dimensional discrete nonpolynomial Schrödinger equation

Two routes to the one-dimensional discrete nonpolynomial Schrödinger equation

Modulated amplitude waves in collisionally inhomogeneous Bose–Einstein condensates

Dynamics of kicked matter-wave solitons in an optical lattice

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