A Generalized-Laguerre–Fourier–Hermite Pseudospectral Method for Computing the Dynamics of Rotating Bose–Einstein Condensates

Abstract

Abstract. A time-splitting generalized-Laguerre-Fourier-Hermite pseudospectral method is proposed for computing the dynamics of rotating Bose-Einstein condensates (BECs) in two and three dimensions. The new numerical method is based on the following: (i) the use of a time-splitting technique for decoupling the nonlinearity; (ii) the adoption of polar coordinate in two dimensions, and resp. cylindrical coordinate in three dimensions, such that the angular rotation term becomes constant coefficient; and (iii) the construction of eigenfunctions for the linear operator by properly scale the generalized-Laguerre, Fourier and Hermite functions. The new method enjoys the following properties: (i) it is explicit, time reversible and time transverse invariant; (ii) it conserves the position density and is spectrally accurate in space and second-order or fourth-order accurate in time; and (iii) it solves the problem in the original whole space instead of in a truncated artificial computational domain. The method is also extended to solve the coupled Gross-Pitaevskii equations for the dynamics of rotating two-component and spin-1 BECs. Extensive numerical results for the dynamics of BECs are reported to demonstrate the accuracy and efficiency of the new method for rotating BECs.

Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation

“…In addition, when the computational domain is a disk in 2D, and resp., a cylinder in 3D, the SIFD discretization can be extremely efficient in practical computation by using polar coordinates in 2D, and resp., cylindrical coordinates in 3D, together with fast direct Poisson solver. A similar idea to this method has been used in simulating quantized vortex dynamics in rotating BEC [6,9,12].…”

“…To our knowledge, no error estimates are available in the literature of finite difference methods for NLS either in high dimensions or for a non-conservative scheme. However, the GPE with the angular momentum rotation is either in 2D or 3D [6,9,11,35]. The main aim of this paper is to use different techniques to establish optimal error bounds of CNFD and the semi-implicit finite difference (SIFD) method for the GPE (1.1) with the angular momentum rotation in 2D and 3D.…”

“…In fact, it is easy to show that the GPE (1.1) conserves the total mass wheref denotes the conjugate of f . Along the numerical front, different efficient and accurate numerical methods including the time-splitting pseudospectral method [7,25,36,37], finite difference method [2,3], and Runge-Kutta or CrankNicolson pseudospectral method [14,20] have been developed for the GPE without and with [6,9,11] the angular momentum rotation term. Of course, each method has its advantages and disadvantages.…”

A review of numerical methods for nonlinear partial differential equations

“…Observe that the computation of the Laplacian, Δ x W j = − |k |≤N |k| 2 W k e ik·x j , is carried out in the Figure 3.10. Contour plots of the density, |w(x, t)| 2 , using a pseudo-spectral computation [13], for the interaction of two vortex dipoles in a rotating two-dimensional Bose-Einstein condensate governed by the Schrödinger equation (2.6) with potential…”

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

“…Other choice of operators A and B (e.g. including a part of the potential V in A) lead to different spectral basis that diagonalize the operators (see [26,28,29] for Hermite or Laguerre polynomials).…”