Abstract:
The achievement of Bose-Einstein condensation (BEC) in ultracold vapors of alkali atoms has given enormous impulse to the theoretical and experimental study of dilute atomic gases in condensed quantum states inside magnetic traps and optical lattices. This article offers a short survey on mathematical models and theories as well as numerical methods for BEC based on the mean field theory. We start with the Gross-Pitaevskii equation (GPE) in three dimensions (3D) for modeling one-component BEC of the weakly interacting bosons, scale it to obtain a three-parameter model and show how to reduce it to two dimensions (2D) and one dimension (1D) GPEs in certain limiting regimes. Mathematical theories and numerical methods for ground states and dynamics of BEC are provided. Extensions to GPE with an angular momentum rotation term for a rotating BEC, to GPE with long-range anisotropic dipole-dipole interaction for a dipolar BEC and to coupled GPEs for spin-orbit coupled BECs are discussed. Finally, some conclusions are drawn and future research perspectives are discussed.

Abstract:
We study numerically the time-independent vector Gross-Pitaevskii equations (VGPEs) for ground states and time-dependent VGPEs with (or without) an external driven field for dynamics describing a multi-component Bose-Einstein condensate (BEC) at zero or very low temperature. In preparation for the numerics, we scale the 3d VGPEs, approximately reduce it to lower dimensions, present a normalized gradient flow (NGF) to compute ground states of multi-component BEC, prove energy diminishing of the NGF which provides a mathematical justification, discretize it by the backward Euler finite difference (BEFD) which is monotone in linear and nonlinear cases and preserves energy diminishing property in linear case. Then we use a time-splitting sine-spectral method (TSSP) to discretize the time-dependent VGPEs with an external driven field for computing dynamics of multi-component BEC. The merit of the TSSP for VGPEs is that it is explicit, unconditionally stable, time reversible and time transverse invariant if the VGPEs is, `good' resolution in the semiclassical regime, of spectral order accuracy in space and second order accuracy in time, and conserves the total particle number in the discretized level. Extensive numerical examples in 3d for ground states and dynamics of multi-component BEC are presented to demonstrate the power of the numerical methods and to discuss the physics of multi-component Bose-Einstein condensates.

Abstract:
In this paper, we propose a new numerical method to compute the ground state solution of trapped interacting Bose-Einstein condensation (BEC) at zero or very low temperature by directly minimizing the energy functional via finite element approximation. As preparatory steps we begin with the 3d Gross-Pitaevskii equation (GPE), scale it to get a three-parameter model and show how to reduce it to 2d and 1d GPEs. The ground state solution is formulated by minimizing the energy functional under a constraint, which is discretized by the finite element method. The finite element approximation for 1d, 2d with radial symmetry and 3d with spherical symmetry and cylindrical symmetry are presented in detail and approximate ground state solutions, which are used as initial guess in our practical numerical computation of the minimization problem, of the GPE in two extreme regimes: very weak interactions and strong repulsive interactions are provided. Numerical results in 1d, 2d with radial symmetry and 3d with spherical symmetry and cylindrical symmetry for atoms ranging up to millions in the condensation are reported to demonstrate the novel numerical method. Furthermore, comparisons between the ground state solutions and their Thomas-Fermi approximations are also reported. Extension of the numerical method to compute the excited states of GPE is also presented.

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 Bose-Einstein 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 time-splitting sine-spectral (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. Crank-Nicolson 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 far-blue 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.

Abstract:
A fourth-order time-splitting Laguerre-Hermite pseudospectral method is introduced for Bose-Einstein condensates (BEC) in 3-D with cylindrical symmetry. The method is explicit, unconditionally stable, time reversible and time transverse invariant. It conserves the position density, and is spectral accurate in space and fourth-order accurate in time. Moreover, the new method has two other important advantages: (i) it reduces a 3-D problem with cylindrical symmetry to an effective 2-D problem; (ii) it solves the problem in the whole space instead of in a truncated artificial computational domain. The method is applied to vector Gross-Pitaevskii equations (VGPEs) for multi-component BECs. Extensive numerical tests are presented for 1-D GPE, 2-D GPE with radial symmetry, 3-D GPE with cylindrical symmetry as well as 3-D VGPEs for two-component BECs to show the efficiency and accuracy of the new numerical method.

Abstract:
We study asymptotically and numerically fundamental gaps (i.e. difference between the first excited state and the ground state) in energy and chemical potential of the Gross-Pitaevskii equation (GPE) -- nonlinear Schr\"{o}dinger equation (NLSE) 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 quadratic in the far field. We extend these results to the GPE on bounded domains with either the homogeneous Neumann boundary condition or periodic boundary condition. Finally, we obtain ground state approximations and their corresponding energy and chemical potential asymptotics of the NLSE with general nonlinearity and different trapping potentials for different parameter regimes and observe bifurcations in the ground states when the power of the nonlinearity goes to $\infty$. Numerical results are reported to support the asymptotic results.

Abstract:
In this paper, we propose an efficient and spectrally accurate numerical method for computing the dynamics of rotating Bose-Einstein condensates (BEC) in two dimensions (2D) and 3D based on the Gross-Pitaevskii equation (GPE) with an angular momentum rotation term. By applying a time-splitting technique for decoupling the nonlinearity and properly using the alternating direction implicit (ADI) technique for the coupling in the angular momentum rotation term in the GPE, at every time step, the GPE in rotational frame is decoupled into a nonlinear ordinary differential equation (ODE) and two partial differential equations with constant coefficients. This allows us to develop new time-splitting spectral (TSSP) methods for computing the dynamics of BEC in a rotational frame.The new numerical method is explicit, unconditionally stable, and of spectral accuracy in space and second order accuracy in time. Moreover, it is time reversible and time transverse invariant, and conserves the position density in the discretized level if the GPE does. Extensive numerical results are presented to confirm the above properties of the new numerical method for rotating BEC in 2D and 3D.

Abstract:
We study analytically and asymptotically as well as numerically ground states and dynamics of two-component spin-orbit-coupled Bose-Einstein condensates (BECs) modeled by the coupled Gross-Pitaevskii equations (CGPEs). In fact, due to the appearance of the spin-orbit (SO) coupling in the two-component BEC with a Raman coupling, the ground state structures and dynamical properties become very rich and complicated. For the ground states, we establish the existence and non-existence results under different parameter regimes, and obtain their limiting behaviors and/or structures with different combinations of the SO and Raman coupling strengths. For the dynamics, we show that the motion of the center-of-mass is either non-periodic or with different frequency to the trapping frequency when the external trapping potential is taken as harmonic and the initial data is chosen as a stationary state (e.g. ground state) with a shift, which is completely different from the case of a two-component BEC without the SO coupling, and obtain the semiclassical limit of the CGPEs in the linear case via the Wigner transform method. Efficient and accurate numerical methods are proposed for computing the ground states and dynamics, especially for the case of box potentials. Numerical results are reported to demonstrate the efficiency and accuracy of the numerical methods and show the rich phenomenon in the SO-coupled BECs.

Abstract:
In this paper, we mainly review recent results on mathematical theory and numerical methods for Bose-Einstein condensation (BEC), based on the Gross-Pitaevskii equation (GPE). Starting from the simplest case with one-component 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, well-posedness of the Cauchy problem as well as the finite time blow-up. 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 finite difference methods and time splitting spectral methods are reviewed, 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 (Thomas-Fermi regime), and discuss semiclassical limits of the GPE. Extensions of these results for one-component BEC are then carried out for rotating BEC by GPE with an angular momentum rotation, dipolar BEC by GPE with long range dipole-dipole interaction, and two-component BEC by coupled GPEs. Finally, as a perspective, we show briefly the mathematical models for spin-1 BEC, Bogoliubov excitation and BEC at finite temperature.

Abstract:
We propose a framework for calculating scattering and bound state properties in anisotropic two-dimensional potentials. Using our method, we derive systematic approximations of partial wave phase shifts and binding energies. Moreover, the method is suitable for efficient numerical computations. We calculate the s-wave phase shift and binding energy of polar molecules in two layers polarized by an external field along an arbitrary direction. We find that scattering depends strongly on their polarization direction and that absolute interlayer binding energies are larger than thermal energies at typical ultracold temperatures.