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1. Error estimates of local energy regularization for the logarithmic Schrodinger equation

Author

Chunmei Su, Rémi Carles, Qinglin Tang, Weizhu Bao, Department of Mathematics [Singapore], National University of Singapore (NUS), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Yau Mathematical Sciences Center, Tsinghua University, School of Mathematics, State Key Laboratory of Hydraulics and Mountain River Engineering, Chengdu University of Technology (CDUT), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-Centre National de la Recherche Scientifique (CNRS), and Technische Universität München [München] (TUM)

Subjects
Quadratic growth, Polynomial, Partial differential equation, Logarithm, Applied Mathematics, 010102 general mathematics, Numerical Analysis (math.NA), 16. Peace & justice, 01 natural sciences, Regularization (mathematics), 010101 applied mathematics, Nonlinear system, Rate of convergence, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Modeling and Simulation, FOS: Mathematics, Applied mathematics, Mathematics - Numerical Analysis, 0101 mathematics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics, Energy functional
Abstract

The logarithmic nonlinearity has been used in many partial differential equations (PDEs) for modeling problems in various applications.Due to the singularity of the logarithmic function, it introducestremendous difficulties in establishing mathematical theories, as well asin designing and analyzing numerical methods for PDEs with such nonlinearity. Here we take the logarithmic Schr\"odinger equation (LogSE)as a prototype model. Instead of regularizing $f(\rho)=\ln \rho$ in theLogSE directly and globally as being done in the literature, we propose a local energy regularization (LER) for the LogSE byfirst regularizing $F(\rho)=\rho\ln \rho -\rho$ locally near $\rho=0^+$ with a polynomial approximation in the energy functional of the LogSE and then obtaining an energy regularized logarithmic Schr\"odinger equation (ERLogSE) via energy variation. Linear convergence is established between the solutions of ERLogSE and LogSE in terms of a small regularization parameter $0, Comment: 31 pages, 10 figures, final version. More explanations and some proofs are more detailed

Published
2022
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2. Regularized numerical methods for the logarithmic Schrodinger equation

Author

Weizhu Bao, Chunmei Su, Rémi Carles, Qinglin Tang, Department of Mathematics [Singapore], National University of Singapore (NUS), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Department of Mathematics, University of Innsbruck, School of Mathematics, Sichuan University, Sichuan University [Chengdu] (SCU), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), and Universität Innsbruck [Innsbruck]

Subjects
Logarithm, Discretization, Applied Mathematics, Numerical analysis, Finite difference method, Order (ring theory), Numerical Analysis (math.NA), 010103 numerical & computational mathematics, 16. Peace & justice, 01 natural sciences, 010101 applied mathematics, Computational Mathematics, Nonlinear system, Mathematics - Analysis of PDEs, Integrator, FOS: Mathematics, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, 0101 mathematics, Energy (signal processing), [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP), Mathematics
Abstract

We present and analyze two numerical methods for the logarithmic Schr{\"o}dinger equation (LogSE) consisting of a regularized splitting method and a regularized conservative Crank-Nicolson finite difference method (CNFD). In order to avoid numerical blow-up and/or to suppress round-off error due to the logarithmic nonlinearity in the LogSE, a regularized logarithmic Schr{\"o}dinger equation (RLogSE) with a small regularized parameter 0 < $\epsilon$ $\ll$ 1 is adopted to approximate the LogSE with linear convergence rate O($\epsilon$). Then we use the Lie-Trotter splitting integrator to solve the RLogSE and establish its error bound O($\tau$ 1/2 ln($\epsilon$ --1)) with $\tau$ > 0 the time step, which implies an error bound at O($\epsilon$ + $\tau$ 1/2 ln($\epsilon$ --1)) for the LogSE by the Lie-Trotter splitting method. In addition, the CNFD is also applied to discretize the RLogSE, which conserves the mass and energy in the discretized level. Numerical results are reported to confirm our error bounds and to demonstrate rich and complicated dynamics of the LogSE., Comment: 23 pages, 8 colored figures

Published
2019
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3. Fundamental gaps of the Gross–Pitaevskii equation with repulsive interaction

Author

Weizhu Bao, Xinran Ruan, National University of Singapore (NUS), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), and Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)

Subjects
General Mathematics, FOS: Physical sciences, Space (mathematics), repulsive interaction, 01 natural sciences, symbols.namesake, 0103 physical sciences, 35B40, 35P30, 35Q55, 65N25, Neumann boundary condition, Periodic boundary conditions, ground state, 0101 mathematics, [MATH]Mathematics [math], 010306 general physics, Nonlinear Schrödinger equation, Mathematical Physics, Physics, Condensed Matter::Quantum Gases, fundamental gap, Mathematical Physics (math-ph), first excited state, 010101 applied mathematics, Gross–Pitaevskii equation, Classical mechanics, Dirichlet boundary condition, Bounded function, symbols, energy asymptotics, Ground state, Gross-Pitaevskii equation
Abstract

We study asymptotically and numerically the fundamental gaps (i.e. the difference between the first excited state and the ground state) in energy and chemical potential of the Gross-Pitaevskii equation (GPE) -- nonlinear Schrodinger equation with cubic nonlinearity -- with repulsive interaction under different trapping potentials including box potential and harmonic potential. Based on our asymptotic and numerical results, we formulate a gap conjecture on the fundamental gaps in energy and chemical potential of the GPE on bounded domains with the homogeneous Dirichlet boundary condition, and in the whole space with a convex trapping potential growing at least quadratically in the far field. We then extend these results to the GPE on bounded domains with either the homogeneous Neumann boundary condition or periodic boundary condition., Comment: 26 pages, 16 figures

Published
2018
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4. Dimension reduction for dipolar Bose-Einstein condensates in the strong interaction regime

Author

Weizhu Bao, Florian Méhats, Loïc Le Treust, National University of Singapore (NUS), Invariant Preserving SOlvers (IPSO), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), This work was supported by the Ministry of Education of Singaporegrant R-146-000-196-112 (W.B.), by the ANR-FWF Project Lodiquas ANR-11-IS01-0003 (L.L.T. and F.M.), by the ANR-10-BLAN-0101 Grant (L.L.T.) andby the ANR project Moonrise ANR-14-CE23-0007-01 (F.M.)., ANR-11-IS01-0003,LODIQUAS,modelisation et simulation numerique pour les systemes quantiques de basse dimension.(2011), ANR-10-BLAN-0101,NoNAP,Problèmes non linéaires en physique atomique et nucléaire(2010), ANR-14-CE23-0007,MOONRISE,MOdèles, Oscillations et SchEmas NUmeriques(2014), National University of Singapore ( NUS ), Invariant Preserving SOlvers ( IPSO ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ) -Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ) -Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique ( Inria ), ANR-11-IS01-0003,LODIQUAS,modelisation et simulation numerique pour les systemes quantiques de basse dimension. ( 2011 ), ANR-10-BLAN-0101,NoNAP,Problèmes non linéaires en physique atomique et nucléaire ( 2010 ), ANR-14-CE23-0007,MOONRISE,MOdèles, Oscillations et SchEmas NUmeriques ( 2014 ), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-AGROCAMPUS OUEST, Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-Inria Rennes – Bretagne Atlantique, and ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011)

Subjects
Asymptotic analysis, Semiclassical analysis, Strong interaction, Semiclassical physics, 01 natural sciences, law.invention, Mathematics - Analysis of PDEs, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Dimension (vector space), law, Quantum mechanics, 0103 physical sciences, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, 010306 general physics, Physics, Condensed Matter::Quantum Gases, Numerical Analysis, Partial differential equation, Condensed Matter::Other, Partial Differential Equations, 010101 applied mathematics, Gross–Pitaevskii equation, Dipolar Bose-Einstein Condensates, Averaging techniques, Modeling and Simulation, 35Q55, 35Q40, Bose–Einstein condensate, Analysis of PDEs (math.AP), Dimensionless quantity
Abstract

International audience; We study dimension reduction for the three-dimensional Gross-Pitaevskii equation with a long-range and anisotropic dipole-dipole interaction modeling dipolar Bose-Einstein condensation in a strong interaction regime. The cases of disk shaped condensates (confinement from dimension three to dimension two) and cigar shaped condensates (confinement to dimension one) are analyzed. In both cases, the analysis combines averaging tools and semiclassical techniques. Asymptotic models are derived, with rates of convergence in terms of two small dimensionless parameters characterizing the strength of the confinement and the strength of the interaction between atoms.

Published
2017
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5. Computing the ground state and dynamics of the nonlinear Schrödinger equation with nonlocal interactions via the nonuniform FFT

Author

Shidong Jiang, Weizhu Bao, Yong Zhang, Qinglin Tang, Department of Mathematics [Singapore], National University of Singapore ( NUS ), Department of Mathematical Sciences [Newark, NJ] ( NJIT ), State University of New Jersey, Equations aux dérivées partielles ( EDP ), Institut Élie Cartan de Lorraine ( IECL ), Université de Lorraine ( UL ) -Centre National de la Recherche Scientifique ( CNRS ) -Université de Lorraine ( UL ) -Centre National de la Recherche Scientifique ( CNRS ), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization ( SPHINX ), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ), Wolfgang Pauli Institute ( WPI ), University of Vienna [Vienna], Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), ANR-12-MONU-0007,BECASIM,Simulation numérique avancée pour les condensats de Bose-Einstein : Modèles numériques déterministes et stochastiques, Calcul haute performance, Simulation d'expériences physiques ( 2012 ), National University of Singapore (NUS), Department of Mathematical Sciences [Newark, NJ] (NJIT), Rutgers, The State University of New Jersey [New Brunswick] (RU), Rutgers University System (Rutgers)-Rutgers University System (Rutgers), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Wolfgang Pauli Institute (WPI), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), R-146-000-196-112, Ministry of Education - Singapore, DMS-1418918, National Science Foundation, 12-MONU-0007-02, ANR, I830, Austrian Science Fund, 91430103, Natural Science Foundation of China, Austrian Ministry of Science and Research, ANR-12-MONU-0007,BECASIM,Simulation numérique avancée pour les condensats de Bose-Einstein : Modèles numériques déterministes et stochastiques, Calcul haute performance, Simulation d'expériences physiques(2012), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), and ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011)

Subjects
Numerical Analysis, Physics and Astronomy (miscellaneous), Discretization, Applied Mathematics, Numerical analysis, Mathematical analysis, Fast Fourier transform, [ MATH.MATH-MP ] Mathematics [math]/Mathematical Physics [math-ph], Numerical Analysis (math.NA), [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], Backward Euler method, Computer Science Applications, Split-step method, Computational Mathematics, symbols.namesake, Singularity, Fourier transform, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Modeling and Simulation, symbols, FOS: Mathematics, Mathematics - Numerical Analysis, Nonlinear Schrödinger equation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

We present efficient and accurate numerical methods for computing the ground state and dynamics of the nonlinear Schr\"odinger equation (NLSE) with nonlocal interactions based on a fast and accurate evaluation of the long-range interactions via the nonuniform fast Fourier transform (NUFFT). We begin with a review of the fast and accurate NUFFT based method in \cite{JGB} for nonlocal interactions where the singularity of the Fourier symbol of the interaction kernel at the origin can be canceled by switching to spherical or polar coordinates. We then extend the method to compute other nonlocal interactions whose Fourier symbols have stronger singularity at the origin that cannot be canceled by the coordinate transform. Many of these interactions do not decay at infinity in the physical space, which adds another layer of complexity since it is more difficult to impose the correct artificial boundary conditions for the truncated bounded computational domain. The performance of our method against other existing methods is illustrated numerically, with particular attention on the effect of the size of the computational domain in the physical space. Finally, to study the ground state and dynamics of the NLSE, we propose efficient and accurate numerical methods by combining the NUFFT method for potential evaluation with the normalized gradient flow using backward Euler Fourier pseudospectral discretization and time-splitting Fourier pseudospectral method, respectively. Extensive numerical comparisons are carried out between these methods and other existing methods for computing the ground state and dynamics of the NLSE with various nonlocal interactions. Numerical results show that our scheme performs much better than those existing methods in terms of both accuracy and efficiency., Comment: 23 pages, 3 figures

Published
2015
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6. Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations

Author

Weizhu Bao, Christophe Besse, Xavier Antoine, Robust control of infinite dimensional systems and applications (CORIDA), Institut Élie Cartan de Nancy (IECN), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [Singapore], National University of Singapore (NUS), Laboratoire Paul Painlevé - UMR 8524 (LPP), Centre National de la Recherche Scientifique (CNRS)-Université de Lille, SImulations and Modeling for PArticles and Fluids (SIMPAF), Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), This work was partially supported by the French ANR grants MicroWave NT09_460489 (‘‘Programme Blanc’’ call) and ANR-12-MONU-0007-02 BECASIM (‘‘Modèles Numériques’’ call) (X.Antoine and C.Besse), This work was partially supported by the Singapore A*STARSERC ‘‘ComplexSystems’’ Research Programme grant 1224504056 (W. Bao)., ANR-12-MONU-0007,BECASIM,Simulation numérique avancée pour les condensats de Bose-Einstein : Modèles numériques déterministes et stochastiques, Calcul haute performance, Simulation d'expériences physiques(2012), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS), Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Inria Lille - Nord Europe, Equations aux dérivées partielles (EDP), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), and Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Inria Nancy - Grand Est

Subjects
Angular momentum, [PHYS.COND.GAS]Physics [physics]/Condensed Matter [cond-mat]/Quantum Gases [cond-mat.quant-gas], General Physics and Astronomy, Bose-Einstein condensation, 01 natural sciences, 7. Clean energy, 010305 fluids & plasmas, law.invention, symbols.namesake, [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], law, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, Boundary value problem, 35Q41, 65M06, 65M12, 65M70, 65Z05, 78M20, 78M22, 81Q05, 81Q20, 81V45, 0101 mathematics, nonlinear Schrödinger equation, time-splitting spectral method, Nonlinear Schrödinger equation, Nonlinear Sciences::Pattern Formation and Solitons, ComputingMilieux_MISCELLANEOUS, Mathematical physics, Physics, Condensed Matter::Quantum Gases, Condensed Matter::Other, Numerical analysis, Numerical Analysis (math.NA), 010101 applied mathematics, Nonlinear system, Gross–Pitaevskii equation, Classical mechanics, Hardware and Architecture, absorbing boundary condition, symbols, Spectral method, Bose–Einstein condensate, Crank-Nicolson finite difference method, Gross-Pitaevskii equation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

In this paper, we begin with the nonlinear Schrodinger/Gross-Pitaevskii equation (NLSE/GPE) for modeling Bose-Einstein condensation (BEC) and nonlinear optics as well as other applications, and discuss their dynamical properties ranging from time reversible, time transverse invariant, mass and energy conservation, dispersion relation to soliton solutions. Then, we review and compare different numerical methods for solving the NLSE/GPE including finite difference time domain methods and time-splitting spectral method, and discuss different absorbing boundary conditions. In addition, these numerical methods are extended to the NLSE/GPE with damping terms and/or an angular momentum rotation term as well as coupled NLSEs/GPEs. Finally, applications to simulate a quantized vortex lattice dynamics in a rotating BEC are reported., 23 pages 1 figure

Published
2013
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7. Gross-Pitaevskii-Poisson Equations for Dipolar Bose-Einstein Condensate with Anisotropic Confinement

Author

Weizhu Bao, Naoufel Ben Abdallah, Yongyong Cai, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [Singapore], National University of Singapore (NUS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
FOS: Physical sciences, 01 natural sciences, 82Cxx (35Q55), law.invention, 35Q55, 35A01, 81Q99, law, Quantum mechanics, 0103 physical sciences, Initial value problem, Uniqueness, 0101 mathematics, 010306 general physics, Anisotropy, Mathematical Physics, ComputingMilieux_MISCELLANEOUS, Mathematical physics, Physics, Condensed Matter::Quantum Gases, Applied Mathematics, Operator (physics), Mathematical analysis, Mathematical Physics (math-ph), 010101 applied mathematics, Computational Mathematics, Rate of convergence, Poisson's equation, Ground state, Analysis, Bose–Einstein condensate
Abstract

Ground states and dynamical properties of dipolar Bose-Einstein condensate are analyzed based on the Gross-Pitaevskii-Poisson system (GPPS) and its dimension reduction models under anisotropic confining potential. We begin with the three-dimensional (3D) Gross-Pitaevskii-Poisson system and review its quasi-2D approximate equations when the trap is strongly confined in $z$-direction and quasi-1D approximate equations when the trap is strongly confined in $x$-, $y$-directions. In fact, in the quasi-2D equations, a fractional Poisson equation with the operator $(-\Delta)^{1/2}$ is involved which brings significant difficulties into the analysis. Existence and uniqueness as well as nonexistence of the ground state under different parameter regimes are established for the quasi-2D and quasi-1D equations. Well-posedness of the Cauchy problem for both equations and finite time blowup in 2D are analyzed. Finally, we rigorously prove the convergence and linear convergence rate between the solutions of the 3D GPPS and its quasi-2D and quasi-1D approximate equations in weak interaction regime., Comment: 29 pages

Published
2012
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8. Introduction to numerical stochastic homogenization and the related computational challenges: some recent developments

Author

A. Anantharaman, R. Costaouec, C. Le Bris, F. Legoll, F. Thomines, Methods and engineering of multiscale computing from atom to continuum (MICMAC), Inria Paris-Rocquencourt, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-École des Ponts ParisTech (ENPC), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), Modélisation et expérimentation multi-échelle pour les solides hétérogènes (multi-échelle), Laboratoire Navier (navier umr 8205), Institut Français des Sciences et Technologies des Transports, de l'Aménagement et des Réseaux (IFSTTAR)-École des Ponts ParisTech (ENPC)-Centre National de la Recherche Scientifique (CNRS)-Institut Français des Sciences et Technologies des Transports, de l'Aménagement et des Réseaux (IFSTTAR)-École des Ponts ParisTech (ENPC)-Centre National de la Recherche Scientifique (CNRS), and Weizhu Bao and Qiang Du

Subjects
010102 general mathematics, 010103 numerical & computational mathematics, 0101 mathematics, 01 natural sciences, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

ISBN: 978-981-4360-89-0; International audience

Published
2011

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