Modulational instability of Gross-Pitaevskii-type equations in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:math>dimensions

Theocharis, G.; Rapti, Zoi; Kevrekidis, P. G.; Frantzeskakis, D. J.; Konotop, V. V.

doi:10.1103/physreva.67.063610

Cited by 147 publications

(114 citation statements)

References 24 publications

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“…[31], where a slow-light technique was used to produce density depressions in a sodium BEC. More recently, they were observed in rapidly rotating 87 Rb BECs triggered by repulsive laser pulses [395], while their formation was discussed in an experiment involving the growth dynamics of a 1D sodium quasi-condensate in a dimple microtrap created on top of the harmonic confinement of an atom chip [396]. Finally, shock waves were studied experimentally and theoretically in Ref.…”

Section: Shock Wavesmentioning

confidence: 99%

“…Similar types of lens transformations were used to study collapse type phenomena [209,210] and to examine modulational instabilities in the presence of parabolic potentials [87].…”

Section: Direct Methodsmentioning

confidence: 99%

“…Modulational instability is responsible for the formation of bright matter-wave solitons [40][41][42], as was analyzed by various theoretical works (see, e.g., Refs. [87][88][89] and the reviews [90,91]). …”

Section: Small-amplitude Linear Excitationsmentioning

confidence: 99%

“…Subsequently, experiments in optically trapped 23 Na [290] were able to produce multicomponent condensates for different Zeeman sub-levels of the same hyperfine level, the so-called spinor condensates. In addition to these two classes of experiments, mixtures of two different species of condensates have been created by sympathetic cooling (i.e., condensing one species and allowing the other one to condense by taking advantage of the coupling with the first species) were 41 K atoms were sympathetically cooled by 87 Rb atoms [291]. More exotic mixtures are also being currently explored in degenerate fermionboson mixtures in 40 K-87 Rb [292] and pure degenerate fermion mixtures in 40 K- 6 Li [293][294][295].…”

Section: Spinor/multicomponent Condensatesmentioning

confidence: 99%

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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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Section: Shock Wavesmentioning

confidence: 99%

“…Similar types of lens transformations were used to study collapse type phenomena [209,210] and to examine modulational instabilities in the presence of parabolic potentials [87].…”

Section: Direct Methodsmentioning

confidence: 99%

Section: Small-amplitude Linear Excitationsmentioning

confidence: 99%

Section: Spinor/multicomponent Condensatesmentioning

confidence: 99%

See 2 more Smart Citations

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

2008

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“…This transformation is analogous to the so-called lens transformation [18] which has been also used in other BEC problems [19,20] and differs (mainly because of the extra freedom provided by the definition of the new time τ ) from the transformations which are frequent in the context of dispersion-managed solitons [15]. For simplicity and without loss of generality we choose t 0 = 0, ℓ(0) = 1, L(0) = 1 and τ (0) = 0.…”

Section: Model Statement and Equations For The Transformation Parametersmentioning

confidence: 99%

Similarity transformations for nonlinear Schrödinger equations with time-dependent coefficients

Pérez-Garcı́a

Torres

Konotop

2006

Physica D: Nonlinear Phenomena

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In this paper we use a similarity transformation connecting some families of Nonlinear Schrödinger equations with time-varying coefficients with the autonomous cubic nonlinear Schrödinger equation. This transformation allows one to apply all known results for that equation to the original non-autonomous case with the additional dynamics introduced by the transformation itself. In particular, using stationary solutions of the autonomous nonlinear Schrödinger equation we can construct exact breathing solutions to multidimensional non-autonomous nonlinear Schrödinger equations. An application is given in which we explicitly construct time dependent coefficients leading to solutions displaying weak collapse in three-dimensional scenarios. Our results can find physical applicability in mean field models of Bose-Einstein condensates and in the field of dispersion-managed optical systems.

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Modulation of ion sound excitations in electron–ion plasmas with double spectral index distribution function

Ullah

Masood

Siddiq

2020

Contributions to Plasma Physics

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Theoretical investigation of amplitude modulation of ion sound waves is presented here for an electron–ion plasma where the electrons are dictated by the double spectral index (r, q) distribution function. Using the standard reductive perturbative technique, a non‐linear Schrödinger (NLS) equation is derived that describes the evolution of modulated ion sound envelope excitations. Stability analysis of the NLS equation shows that the ion sound waves remain stable for the flat‐topped and kappa‐like distributions, but they can become unstable for the spiky electron velocity distribution. It is shown that changing the electron population in regions of low and high phase space density regions results in remarkable features that have no equivalent in ion sound waves with Boltzmannian electrons. Different types of localized ion sound excitations are plotted for the different shapes of the distribution functions controlled by the double spectral indices, and the underlying physics is discussed in detail. The present investigation may be beneficial to understand ion sound excitations in space plasmas where the distribution functions of the shapes presented here are frequently encountered by the satellite missions.

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Modulational instability of Gross-Pitaevskii-type equations in1+1dimensions

Cited by 147 publications

References 24 publications

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

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

Similarity transformations for nonlinear Schrödinger equations with time-dependent coefficients

Modulation of ion sound excitations in electron–ion plasmas with double spectral index distribution function

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