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139
Numerical solution of the GrossPitaevskii equation for BoseEinstein condensation
 J. Comput. Phys
"... We study the numerical solution of the timedependent GrossPitaevskii equation (GPE) describing a BoseEinstein condensate (BEC) at zero or very low temperature. In preparation for the numerics we scale the 3d GrossPitaevskii equation and obtain a fourparameter model. Identifying ‘extreme paramet ..."
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Cited by 111 (56 self)
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We study the numerical solution of the timedependent GrossPitaevskii equation (GPE) describing a BoseEinstein condensate (BEC) at zero or very low temperature. In preparation for the numerics we scale the 3d GrossPitaevskii equation and obtain a fourparameter model. Identifying ‘extreme parameter regimes’, the model is accessible to analytical perturbation theory, which justifies formal procedures well known in the physical literature: reduction to 2d and 1d GPEs, approximation of ground state solutions of the GPE and geometrical optics approximations. Then we use a timesplitting spectral method to discretize the timedependent GPE. Again, perturbation theory is used to understand the discretization scheme and to choose the spatial/temporal grid in dependence of the perturbation parameter. Extensive numerical examples in 1d, 2d and 3d for weak/strong interactions, defocusing/focusing nonlinearity, and zero/nonzero initial phase data are presented to demonstrate the power of the numerical method and to discuss the physics of BoseEinstein condensation.
Numerical study of timesplitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimes
 SIAM J. SCI. COMPUT
, 2003
"... In this paper we study the performance of timesplitting spectral approximations for general nonlinear Schrödinger equations (NLS) in the semiclassical regimes, where the Planck constant ε is small. The timesplitting spectral approximation under study is explicit, unconditionally stable and conse ..."
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Cited by 91 (43 self)
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In this paper we study the performance of timesplitting spectral approximations for general nonlinear Schrödinger equations (NLS) in the semiclassical regimes, where the Planck constant ε is small. The timesplitting spectral approximation under study is explicit, unconditionally stable and conserves the position density in L 1. Moreover it is timetransverse invariant and timereversible when the corresponding NLS is. Extensive numerical tests are presented for weak/strong focusing/defocusing nonlinearities, for the Gross–Pitaevskii equation, and for currentrelaxed quantum hydrodynamics. The tests are geared towards the understanding of admissible meshing strategies for obtaining “correct” physical observables in the semiclassical regimes. Furthermore, comparisons between the solutions of the NLS and its hydrodynamic semiclassical limit are presented.
Computing the ground state solution of BoseEinstein condensates by a normalized gradient flow
 SIAM J. Sci. Comput
, 2004
"... Abstract. In this paper, we prove the energy diminishing of a normalized gradient flow which provides a mathematical justification of the imaginary time method used in physical literatures to compute the ground state solution of BoseEinstein condensates (BEC). We also investigate the energy diminis ..."
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Cited by 88 (28 self)
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Abstract. In this paper, we prove the energy diminishing of a normalized gradient flow which provides a mathematical justification of the imaginary time method used in physical literatures to compute the ground state solution of BoseEinstein condensates (BEC). We also investigate the energy diminishing property for the discretization of the normalized gradient flow. Two numerical methods are proposed for such discretizations: one is the backward Euler centered finite difference (BEFD), the other one is an explicit timesplitting sinespectral (TSSP) method. Energy diminishing for BEFD and TSSP for linear case, and monotonicity for BEFD for both linear and nonlinear cases are proven. Comparison between the two methods and existing methods, e.g. CrankNicolson finite difference (CNFD) or forward Euler finite difference (FEFD), shows that BEFD and TSSP are much better in terms of preserving energy diminishing property of the normalized gradient flow. Numerical results in 1d, 2d and 3d with magnetic trap confinement potential, as well as a potential of a stirrer corresponding to a farblue detuned Gaussian laser beam are reported to demonstrate the effectiveness of BEFD and TSSP methods. Furthermore we observe that the normalized gradient flow can also be applied directly to compute the first excited state solution in BEC when the initial data is chosen as an odd function.
Ground states and dynamics of multicomponent BoseEinstein condensates. Multiscale Modeling and Simulation
, 2004
"... Abstract. We study numerically the timeindependent vector Gross–Pitaevskii equations (VGPEs) for ground states and timedependent VGPEs with (or without) an external driven field for dynamics describing a multicomponent Bose–Einstein condensate (BEC) at zero or a very low temperature. In preparatio ..."
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Cited by 59 (27 self)
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Abstract. We study numerically the timeindependent vector Gross–Pitaevskii equations (VGPEs) for ground states and timedependent VGPEs with (or without) an external driven field for dynamics describing a multicomponent Bose–Einstein condensate (BEC) at zero or a very low temperature. In preparation for the numerics, we scale the threedimensional (3d) VGPEs, approximately reduce it to lower dimensions, present a continuous normalized gradient flow (CNGF) to compute ground states of multicomponent BEC, prove energy diminishing of the CNGF, which provides a mathematical justification, and discretize it by the backward Euler finite difference (BEFD), which is monotone in linear and nonlinear cases and preserves energy diminishing property in the linear case. Then we use a timesplitting sinespectral (TSSP) method to discretize the timedependent VGPEs with an external driven field for computing dynamics of multicomponent BEC. The merits of the TSSP method for VGPEs are that it is explicit, unconditionally stable, time reversible and time transverse invariant if the VGPEs is, has “good ” resolution in the semiclassical regime, is of spectralorder accuracy in space and secondorder accuracy in time, and conserves the total particle number in the discretized level. Extensive numerical examples in three dimensions for ground states and dynamics of multicomponent BEC are presented to demonstrate the power of the numerical methods and to discuss the physics of multicomponent BEC.
MultiPhase Computations Of The Semiclassical Limit Of The Schrödinger Equation And Related Problems: Whitham Vs Wigner
 Wigner, Physica D
"... We present and compare two different techniques to obtain the multiphase solutions for the Schrödinger equation in the semiclassical limit. The first is Whitham's averaging method, which gives the modulation equations governing the evolution of multiphase solutions. The second is the Wigner t ..."
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Cited by 52 (17 self)
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We present and compare two different techniques to obtain the multiphase solutions for the Schrödinger equation in the semiclassical limit. The first is Whitham's averaging method, which gives the modulation equations governing the evolution of multiphase solutions. The second is the Wigner transform, a convenient tool to derive the semiclassical limit equation in the phase space (the Vlasov equation) for the linear Schrödinger equation. Motivated by the linear superposition principle, we derive and prove the multiphase ansatz for the Wigner function by the stationary phase method, and then use the ansatz to close the moment equations of the Vlasov equation and obtain the multiphase equations in the physical space. We show that the multiphase equations so derived agree with those derived by Whitham's averaging method, which can be proved using different arguments. Generic way of obtaining and computing the multiphase equations by the Wigner function is given, and kinetic schemes are introduced to solve the multiphase equations. The numerical schemes are purely Eulerian and only operate in the physical space. Several numerical examples are given to explore the validity of this approach. Similar studies are conducted for the linearized Kortewegde Vries equation and the linear wave equation.
A fourthorder timesplitting LaguerreHermite pseudospectral method for BoseEinstein condensates
 SIAM JOURNAL ON SCIENTIFIC COMPUTING
, 2005
"... A fourthorder timesplitting LaguerreHermite pseudospectral method is introduced for BoseEinstein condensates (BECs) in three dimensions with cylindrical symmetry. The method is explicit, time reversible, and time transverse invariant. It conserves the position density and is spectral accurate ..."
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Cited by 45 (16 self)
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A fourthorder timesplitting LaguerreHermite pseudospectral method is introduced for BoseEinstein condensates (BECs) in three dimensions with cylindrical symmetry. The method is explicit, time reversible, and time transverse invariant. It conserves the position density and is spectral accurate in space and fourthorder accurate in time. Moreover, the new method has two other important advantages: (i) it reduces a threedimensional (3D) problem with cylindrical symmetry to an effective twodimensional (2D) problem; (ii) it solves the problem in the whole space instead of in a truncated artificial computational domain. The method is applied to vector GrossPitaevskii equations (VGPEs) for multicomponent BECs. Extensive numerical tests are presented for the onedimensional (1D) GPE, the 2D GPE with radial symmetry, the 3D GPE with cylindrical symmetry, as well as 3D VGPEs for twocomponent BECs, to show the efficiency and accuracy of the new numerical method.
An explicit unconditionally stable numerical method for solving damped nonlinear Schrödinger equations with a focusing nonlinearity
 SIAM J. Numer. Anal
"... Abstract. This paper introduces an extension of the timesplitting sinespectral (TSSP) method for solving damped focusing nonlinear Schrödinger equations (NLSs). The method is explicit, unconditionally stable, and time transversal invariant. Moreover, it preserves the exact decay rate for the norma ..."
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Cited by 43 (25 self)
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Abstract. This paper introduces an extension of the timesplitting sinespectral (TSSP) method for solving damped focusing nonlinear Schrödinger equations (NLSs). The method is explicit, unconditionally stable, and time transversal invariant. Moreover, it preserves the exact decay rate for the normalization of the wave function if linear damping terms are added to the NLS. Extensive numerical tests are presented for cubic focusing NLSs in two dimensions with a linear, cubic, or quintic damping term. Our numerical results show that quintic or cubic damping always arrests blowup, while linear damping can arrest blowup only when the damping parameter δ is larger than a threshold value δth. We note that our method can also be applied to solve the threedimensional Gross–Pitaevskii equation with a quintic damping term to model the dynamics of a collapsing and exploding Bose–Einstein condensate (BEC). Key words. damped nonlinear Schrödinger equation (DNLS), timesplitting sinespectral
Multiphase Semiclassical Approximation of an Electron in a OneDimensional Crystalline Lattice I. Homogeneous Problems
, 2003
"... We present a computational approach for the WKB approximation of the wave function of an electron moving in a periodic onedimensional crystal lattice. We derive a nonstrictly hyperbolic system for the phase and the intensity where the flux functions originate from the Bloch spectrum of the Schrodi ..."
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Cited by 36 (8 self)
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We present a computational approach for the WKB approximation of the wave function of an electron moving in a periodic onedimensional crystal lattice. We derive a nonstrictly hyperbolic system for the phase and the intensity where the flux functions originate from the Bloch spectrum of the Schrodinger operator. Relying on the framework of the multibranch entropy solutions introduced by Brenier and Corrias, we compute e#ciently multiphase solutions using well adapted and simple numerical schemes. In this first part we present computational results for vanishing exterior potentials which demonstrate the e#ectiveness of the proposed method.
Fourth order timestepping for low dispersion Kortewegde Vries and nonlinear Schrödinger equation
 27 T.P. Liu, Development of singularities in the
, 2008
"... Abstract. Purely dispersive equations, such as the Kortewegde Vries and the nonlinear Schrödinger equations in the limit of small dispersion, have solutions to Cauchy problems with smooth initial data which develop a zone of rapid modulated oscillations in the region where the corresponding dispers ..."
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Cited by 30 (18 self)
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Abstract. Purely dispersive equations, such as the Kortewegde Vries and the nonlinear Schrödinger equations in the limit of small dispersion, have solutions to Cauchy problems with smooth initial data which develop a zone of rapid modulated oscillations in the region where the corresponding dispersionless equations have shocks or blowup. Fourth order timestepping in combination with spectral methods is beneficial to numerically resolve the steep gradients in the oscillatory region. We compare the performance of several fourth order methods for the Kortewegde Vries and the focusing and defocusing nonlinear Schrödinger equations in the small dispersion limit: an exponential timedifferencing fourthorder RungeKutta method as proposed by Cox and Matthews in the implementation by Kassam and Trefethen, integrating factors, timesplitting, Fornberg and Driscoll’s ‘sliders’, and an ODE solver in Matlab.
Mathematical theory and numerical methods for BoseEinstein condensation
 Kinet. Relat. Models
"... Abstract. In this paper, we mainly review recent results on mathematical theory and numerical methods for BoseEinstein condensation (BEC), based on the GrossPitaevskii equation (GPE). Starting from the simplest case with onecomponent BEC of the weakly interacting bosons, we study the reduction of ..."
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Cited by 28 (14 self)
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Abstract. In this paper, we mainly review recent results on mathematical theory and numerical methods for BoseEinstein condensation (BEC), based on the GrossPitaevskii equation (GPE). Starting from the simplest case with onecomponent BEC of the weakly interacting bosons, we study the reduction of GPE to lower dimensions, the ground states of BEC including the existence and uniqueness as well as nonexistence results, and the dynamics of GPE including dynamical laws, wellposedness of the Cauchy problem as well as the finite time blowup. To compute the ground state, the gradient flow with discrete normalization (or imaginary time) method is reviewed and various full discretization methods are presented and compared. To simulate the dynamics, both finitedifference methods andtimesplitting spectralmethods arereviewed, and their error estimates are briefly outlined. When the GPE has symmetric properties, we show how to simplify the numerical methods. Then we compare two widely used scalings, i.e. physical scaling (commonly used) and semiclassical scaling, for BEC in strong repulsive interaction regime (ThomasFermi