Multidimensional solitons in periodic potentials

Baizakov, B. B.; Malomed, Boris A.; Salerno, Mario

doi:10.1209/epl/i2003-00579-4

Cited by 272 publications

(279 citation statements)

References 45 publications

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“…Notice that the above variational equations are very similar to the ones derived in Ref. [32] for the 2D cubic NLS (with exactly the same value of ε 0 ), this being a further confirmation of the analogy between the 1D quintic NLS and the usual 2D GPE. We remark that, since N cr does not dependent on ε, the prediction of the VA is that the optical lattice does not affect collapse.…”

Section: Attractive Interactionssupporting

confidence: 83%

Gap-Townes solitons and localized excitations in low-dimensional Bose-Einstein condensates in optical lattices

2005

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We discuss localized ground states of Bose-Einstein condensates in optical lattices with attractive and repulsive three-body interactions in the framework of a quintic nonlinear Schrödinger equation which extends the Gross-Pitaevskii equation to the one dimensional case. We use both a variational method and a self-consistent approach to show the existence of unstable localized excitations which are similar to Townes solitons of the cubic nonlinear Schrödinger equation in two dimensions. These solutions are shown to be located in the forbidden zones of the band structure, very close to the band edges, separating decaying states from stable localized ones (gap-solitons) fully characterizing their delocalizing transition. In this context usual gap solitons appear as a mechanism for arresting collapse in low dimensional BEC in optical lattices with attractive real three-body interaction. The influence of the imaginary part of the three-body interaction, leading to dissipative effects on gap solitons and the effect of atoms feeding from the thermal cloud are also discussed. These results may be of interest for both BEC in atomic chip and Tonks-Girardeau gas in optical lattices.

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Section: Attractive Interactionssupporting

confidence: 83%

Gap-Townes solitons and localized excitations in low-dimensional Bose-Einstein condensates in optical lattices

2005

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“…Such methods have been extensively used in examining very diverse features of BECs including collective excitations [191], studying the dynamics of BECs in 1D optical lattices [187], offering insights on the collapse or absence thereof in higher dimensional settings and potentials -see, e.g., Ref. [192] and references therein-, or on the behavior of BECs on space-or time-dependent nonlinearity settings -see e.g., Ref. [193] as an example-.…”

Section: Direct Methodsmentioning

confidence: 99%

“…[192,440,412]. Moreover, stable bright ring solitons carrying topological charge have been theoretically predicted to exist [441,83,442,443,366] but no experiment has yet corroborated these results.…”

Section: 42mentioning

confidence: 99%

Nonlinear waves in Bose–Einstein condensates: physical relevance and mathematical techniques

2008

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Abstract. The aim of the present review is to introduce the reader to some of the physical notions and of the mathematical methods that are relevant to the study of nonlinear waves in Bose-Einstein Condensates (BECs). Upon introducing the general framework, we discuss the prototypical models that are relevant to this setting for different dimensions and different potentials confining the atoms. We analyze some of the model properties and explore their typical wave solutions (plane wave solutions, bright, dark, gap solitons, as well as vortices). We then offer a collection of mathematical methods that can be used to understand the existence, stability and dynamics of nonlinear waves in such BECs, either directly or starting from different types of limits (e.g., the linear or the nonlinear limit, or the discrete limit of the corresponding equation). Finally, we consider some special topics involving more recent developments, and experimental setups in which there is still considerable need for developing mathematical as well as computational tools.PACS numbers: 03.75. Kk, 03.75.Lm, 05.45 • EP: Ermakov-Pinney (Equation)• GP: Gross-Pitaevskii (Equation)• KdV: Korteweg-de Vries (Equation)• LS: Lyapunov-Schmidt (Technique)• MT: Magnetic Trap• NLS: Nonlinear Schrödinger (Equation)• NPSE: Non-polynomial Schrödinger Equation IntroductionThe phenomenon of Bose-Einstein condensation is a quantum phase transition originally predicted by Bose [1] and Einstein [2,3] in 1924. In particular, it was shown that below a critical transition temperature T c , a finite fraction of particles of a boson gas (i.e., whose particles obey the Bose statistics) condenses into the same quantum state, known as the Bose-Einstein condensate (BEC). Although BoseEinstein condensation is known to be a fundamental phenomenon, connected, e.g., to superfluidity in liquid helium and superconductivity in metals (see, e.g., Ref.[4]), BECs were experimentally realized 70 years after their theoretical prediction: this major achievement took place in 1995, when different species of dilute alkali vapors confined in a magnetic trap (MT) were cooled down to extremely low temperatures [5][6][7], and has already been recognized through the 2001 Nobel prize in Physics [8,9]. This first unambiguous manifestation of a macroscopic quantum state in a manybody system sparked an explosion of activity, as reflected by the publication of several thousand papers related to BECs since then. Nowadays there exist more than fifty experimental BEC groups around the world, while an enormous amount of theoretical work has followed and driven the experimental efforts, with an impressive impact on many branches of Physics.From a theoretical standpoint, and for experimentally relevant conditions, the static and dynamical properties of a BEC can be described by means of an effective mean-field equation known as the Gross-Pitaevskii (GP) equation [10,11]. This is a variant of the famous nonlinear Schrödinger (NLS) equation [12] (incorporating an external potential used to confine th...

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“…In particular, a periodic optical lattice can be produced by counter propagating laser beams. Such potentials can be used to control the soliton parameters [10,11] or to generate gap bright solitons [5]. Propagation in a random linear potential was considered from the point of view of the Anderson localization in BEC and the observation of the crossover between the Anderson localization regime and the nonlinear regime [12].…”

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confidence: 99%

Propagation of matter-wave solitons in periodic and random nonlinear potentials

2005

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We study the motion of bright matter wave solitons in nonlinear potentials, produced by periodic or random spatial variations of the atomic scattering length. We obtain analytical results for the soliton motion, the radiation of matter wave, and the radiative soliton decay in such configurations of the Bose-Einstein condensate. The stable regimes of propagation are analyzed. The results are in remarkable agreement with the numerical simulations of the Gross-Pitaevskii equation with periodic or random spatial variations of the mean field interactions.

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Multidimensional solitons in periodic potentials

Abstract: The existence of stable solitons in two-and three-dimensional (2D and Typeset using REVT E X 1

Cited by 272 publications

References 45 publications

Gap-Townes solitons and localized excitations in low-dimensional Bose-Einstein condensates in optical lattices

Gap-Townes solitons and localized excitations in low-dimensional Bose-Einstein condensates in optical lattices

Nonlinear waves in Bose–Einstein condensates: physical relevance and mathematical techniques

Propagation of matter-wave solitons in periodic and random nonlinear potentials

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