Modeling and Computation of Bose-Einstein Condensates: Stationary States, Nucleation, Dynamics, Stochasticity

Antoine, Xavier; Duboscq, Romain

doi:10.1007/978-3-319-19015-0_2

Cited by 34 publications

(70 citation statements)

References 124 publications

(162 reference statements)

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“…When Ω = 0, we can also reduce the 2d GPE to a 1d GPE by using similar arguments when considering a cigar-shaped condensation [42,70,74,75].…”

Section: The Dimensionless Gpementioning

confidence: 94%

“…imaginary time [42,[57][58][59][60][61][62][63][64]) used to compute the stationary solutions. Examples of potentials and nonlinearities are also provided.…”

Section: Gpelab 1 (Gross-pitaevskii Equation Laboratory) Is a Flexiblementioning

confidence: 99%

“…Rather than a finite difference scheme, pseudospectral approximation of the spatial derivatives can be used to get high-order accuracy in space [42]. GPELab considers an approach based on Fourier series representations through FFTs.…”

Section: Backward Euler Pseudospectral (Besp) Schemementioning

confidence: 99%

“…One of the main purposes of GPELab is to provide to physicists a generic, friendly and robust numerical modeling tool to compute theoretical data related to BECs through GPEs. The numerical methods implemented in GPELab are based on pseudospectral approximation techniques [42][43][44] that lead to highly accurate spatial solutions. In this first paper, we present the GPEs that GPELab can solve, the numerical schemes that are used for computing stationary solutions and the associated GPELab functions.…”

Section: Gpelab 1 (Gross-pitaevskii Equation Laboratory) Is a Flexiblementioning

confidence: 99%

“…Examples of GPELab scripts are fully developed to show some of its possibilities for physics problems of interest. A second paper [45] will present the numerical schemes that are included in GPELab for solving the deterministic [46] and stochastic dynamics of GPEs [24][25][26][27]42], the associated GPELab functions and a few numerical examples. Since the GPEs are Schrödinger-type equations, GPELab could also be used to solve other physics problems that need to numerically compute the solution to a Schrödinger equation like for example in nonlinear laser optics [47][48][49][50].…”

Section: Gpelab 1 (Gross-pitaevskii Equation Laboratory) Is a Flexiblementioning

confidence: 99%

See 4 more Smart Citations

GPELab, a Matlab toolbox to solve Gross–Pitaevskii equations I: Computation of stationary solutions

Antoine

Duboscq

2014

Computer Physics Communications

Self Cite

121

137

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“…When Ω = 0, we can also reduce the 2d GPE to a 1d GPE by using similar arguments when considering a cigar-shaped condensation [42,70,74,75].…”

Section: The Dimensionless Gpementioning

confidence: 94%

“…imaginary time [42,[57][58][59][60][61][62][63][64]) used to compute the stationary solutions. Examples of potentials and nonlinearities are also provided.…”

Section: Gpelab 1 (Gross-pitaevskii Equation Laboratory) Is a Flexiblementioning

confidence: 99%

Section: Backward Euler Pseudospectral (Besp) Schemementioning

confidence: 99%

Section: Gpelab 1 (Gross-pitaevskii Equation Laboratory) Is a Flexiblementioning

confidence: 99%

Section: Gpelab 1 (Gross-pitaevskii Equation Laboratory) Is a Flexiblementioning

confidence: 99%

See 3 more Smart Citations

GPELab, a Matlab toolbox to solve Gross–Pitaevskii equations I: Computation of stationary solutions

Antoine

Duboscq

2014

Computer Physics Communications

Self Cite

121

137

View full text Add to dashboard Cite

Orbital Stability via the Energy–Momentum Method: The Case of Higher Dimensional Symmetry Groups

Bièvre

Nodari

2018

Arch Rational Mech Anal

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We consider the orbital stability of relative equilibria of Hamiltonian dynamical systems on Banach spaces, in the presence of a multi-dimensional invariance group for the dynamics. We prove a persistence result for such relative equilibria, present a generalization of the Vakhitov-Kolokolov slope condition to this higher dimensional setting, and show how it allows to prove the local coercivity of the Lyapunov function, which in turn implies orbital stability. The method is applied to study the orbital stability of relative equilibria of nonlinear Schrödinger and Manakov equations. We provide a comparison of our approach to the one by Grillakis-Shatah-Strauss.

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An analysis of Schwarz waveform relaxation domain decomposition methods for the imaginary-time linear Schrödinger and Gross-Pitaevskii equations

Antoine

Lorin

2017

Numer. Math.

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The aim of this paper is to derive and numerically validate some asymptotic estimates of the convergence rate of Classical and Optimized Schwarz Waveform Relaxation (SWR) domain decomposition methods applied to the computation of the stationary states of the one-dimensional linear and nonlinear Schrödinger equations with a potential. Although SWR methods are currently used for efficiently solving high dimensional partial differential equations, their convergence analysis and most particularly obtaining expressions of their convergence rate remains largely open even in one dimension, except in simple cases. In this work, we tacke this problem for linear and nonlinear one-dimensional Schrödinger equations by developing techniques which can be extended to higher dimensional problems and other types of PDEs. The approach combines the method developed in [24] for the linear advection reaction diffusion equation and the theory of inhomogeneous pseudodifferential operators in conjunction with the associated symbolical asymptotic expansions. For computing the stationary states, we consider the imaginary-time formulation of the Schrödinger equation based on the Continuous Normalized Gradient Flow (CNGF) method and use a semiimplicit Euler scheme for the discretization. Some numerical results in the one-dimensional case illustrate the analysis for both the linear Schrödinger and Gross-Pitaevskii equations.

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Modeling and Computation of Bose-Einstein Condensates: Stationary States, Nucleation, Dynamics, Stochasticity

Cited by 34 publications

References 124 publications

GPELab, a Matlab toolbox to solve Gross–Pitaevskii equations I: Computation of stationary solutions

GPELab, a Matlab toolbox to solve Gross–Pitaevskii equations I: Computation of stationary solutions

Orbital Stability via the Energy–Momentum Method: The Case of Higher Dimensional Symmetry Groups

An analysis of Schwarz waveform relaxation domain decomposition methods for the imaginary-time linear Schrödinger and Gross-Pitaevskii equations

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