Stabilization of solitons of the multidimensional nonlinear Schrödinger equation: matter-wave breathers

Montesinos, Gaspar D.

doi:10.1016/j.physd.2003.12.001

Cited by 132 publications

(132 citation statements)

References 43 publications

(91 reference statements)

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“…[9] is most possibly the reason for the negative result obtained there. More recently, Montesinos et al [10] have also confirmed the conclusion of Ref. [9] that no stable three-dimensional soliton could be obtained by a variation of Kerr nonlinearity.…”

Section: Numerical Calculationmentioning

confidence: 62%

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Stabilization of a light bullet in a layered Kerr medium with sign-changing nonlinearity

Adhikari

2004

Phys. Rev. E

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Section: Numerical Calculationmentioning

confidence: 62%

“…The stabilization of such BEC solitons in two [9,10,13] and three [15] dimensions is already under investigation.…”

Section: Discussionmentioning

confidence: 99%

Stabilization of a light bullet in a layered Kerr medium with sign-changing nonlinearity

Adhikari

2004

Phys. Rev. E

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“…On the contrary, in the absence of a tight transversal trapping, higher dimensional extensions of 1D solitons are generally unstable [12]. However, by restricting the transverse direction(s) of the condensate, it is possible to obtain higher † † This technique was suggested in earlier works in various important applications, as e.g., a means to prevent collapse of higher-dimensional attractive BECs [193,203,410], to produce periodic waves [107], robust matter-wave breathers [241,242,411,412], and so on. dimensional soliton solutions that are stabilized for times longer than the lifespan of the experiments [414][415][416].…”

Section: Multidimensional Solitons and Collapsementioning

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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“…Spatial and temporal periodic changes of the scattering lengths have been shown to be effective tools to change stability properties of nonlinear excitations, leading to the existence of two dimensional bright solitons in one and two component attractive condensates [9][10][11][12][13]. Moreover, they were used to induce long lived Bloch oscillations [14][15][16], dynamical localization [17], Rabi oscillations [18] of BEC gap solitons in optical lattices, Faraday waves [19,20] etc.…”

Section: Introductionmentioning

confidence: 99%

Compacton matter waves in binary Bose gases under strong nonlinear management

et al. 2014

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The existence of compacton matter waves in binary mixtures of quasi one-dimensional BoseEinstein condensates in deep optical lattices and in the presence of nonlinearity management, is first demonstrated. For this, we derive an averaged vector discrete nonlinear Schrödinger equation (DNLSE) and show that compacton solutions of different types can exist as stable excitations. Stability properties are studied by linear analysis and by direct numerical integrations of the DNLSE system and their dependence on the inter-and intra-species scattering lengths, investigated. We show that under proper management conditions, compactons can be very robust excitations that can emerge spontaneously from generic initial conditions. A possible experimental setting for compacton observation is also discussed.

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Stabilization of solitons of the multidimensional nonlinear Schrödinger equation: matter-wave breathers

Cited by 132 publications

References 43 publications

Stabilization of a light bullet in a layered Kerr medium with sign-changing nonlinearity

Stabilization of a light bullet in a layered Kerr medium with sign-changing nonlinearity

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

Compacton matter waves in binary Bose gases under strong nonlinear management

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