The Nonlinear Schrödinger Equation with a Strongly Anisotropic Harmonic Potential

Abdallah, Naoufel Ben; Méhats, Florian; Schmeiser, Christian; Weishäupl, Rada-Maria

doi:10.1137/040614554

Cited by 53 publications

(128 citation statements)

References 16 publications

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“…It is also worth mentioning that more recently a rigorous derivation of the 1D GP equation was presented in Ref. [108], using energy and Strichartz estimates, as well as two anisotropic Sobolev inequalities.…”

Section: Lower-dimensional Gp Equationsmentioning

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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Section: Lower-dimensional Gp Equationsmentioning

confidence: 99%

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

2008

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“…Equation (1.1) also comes from Quantum Mechanics: indeed, the cubic Schrödinger equation in R 3 is now commonly used in the theory of Bose-Einstein condensates and in the theory of superfluidity (see [12] for a mathematical justification). If a confining potential in normal directions to some surface of R 3 is added, the asymptotics are expected to be given by equation (1.1) (see [1]). …”

Section: Introduction Let (Mmentioning

confidence: 99%

High frequency solutions of the nonlinear Schrödinger equation on surfaces

Burq

Gérard²,

Tzvetkov

2009

Quart. Appl. Math.

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Abstract. We address the problem of describing solutions of the nonlinear Schrödin-ger equation on a compact surface in the high frequency regime. In this context, we introduce a nonnegative threshold, depending on the geometry of the surface, which can be seen as a measurement of the nonlinear character of the equation, and we compute this number for the torus and for the sphere, as a consequence of earlier arguments. The last part is devoted to the study, on the sphere, of the critical regime associated to this threshold. We prove that the effective dynamics are described by a new evolution equation, the Resonant Hermite-Schrödinger equation.

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“…For bosons, especially Bose-Einstein condensation (BEC) [2,41], by using a Hartree ansatz, the linear Schrödinger equation for N bosons under the short-range Fermi (or contact) interaction is well approximated by the Gross-Pitaevskii equation (GPE) in 3D which is an NLS equation with cubic nonlinearity [9,20,41]. Rigorous mathematical justification for this reduction can be found in the literature [5,28,35,36,37] for the ground state and dynamics of BEC.…”

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confidence: 99%

“…The dimension reduction results from either a geometrical symmetry (e.g., a translational invariance in 1D or 2D) or confining the quantum particles in either one dimension (e.g., 2D "electron sheets") or two dimensions (e.g., 1D "quantum wires") or even all three space dimensions (0D "quantum dots") [16,34,43]. In fact, the confinement can be modeled by adding to the Hamiltonian operator an exterior confining potential with a small parameter, e.g., an anisotropic harmonic oscillator potential [17,19,20]. The small parameter limit of infinitely strong confinement then yields the correct asymptotic model in lower dimensions.…”

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confidence: 99%

Dimension Reduction of the Schrödinger Equation with Coulomb and Anisotropic Confining Potentials

Bao¹,

Jian²,

Mauser³

et al. 2013

SIAM J. Appl. Math.

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Abstract. We consider dimension reduction for the three-dimensional (3D) Schrödinger equation with the Coulomb interaction and an anisotropic confining potential to lower-dimensional models in the limit of infinitely strong confinement in one or two space dimensions and obtain formally the surface adiabatic model (SAM) or surface density model (SDM) in two dimensions (2D) and the line adiabatic model (LAM) in one dimension (1D). Efficient and accurate numerical methods for computing ground states and dynamics of the SAM, SDM, and LAM models are presented based on efficient and accurate numerical schemes for evaluating the effective potential in lower-dimensional models. They are applied to find numerically convergence and convergence rates for the dimension reduction from 3D to 2D and 3D to 1D in terms of ground state and dynamics, which confirm some existing analytical results for the dimension reduction in the literature. In particular, we explain and demonstrate that the standard Schrödinger-Poisson system in 2D is not appropriate to simulate a "2D electron gas" of point particles confined to a plane (or, more generally, a 2D manifold), whereas SDM should be the correct model to be used for describing the Coulomb interaction in 2D in which the square root of Laplacian operator is used instead of the Laplacian operator. Finally, we report ground states and dynamics of the SAM and SDM in 2D and LAM in 1D under different setups.

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The Nonlinear Schrödinger Equation with a Strongly Anisotropic Harmonic Potential

Cited by 53 publications

References 16 publications

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

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

High frequency solutions of the nonlinear Schrödinger equation on surfaces

Dimension Reduction of the Schrödinger Equation with Coulomb and Anisotropic Confining Potentials

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