A Time-Splitting and Sine Spectral Method for Dynamics of Dipolar Bose-Einstein Condensate

Abstract

A two-component Bose-Einstein condensate (BEC) described by two coupled a three-dimension Gross-Pitaevskii (GP) equations is considered, where one equation has dipole-dipole interaction while the other one has only the usual s-wave contact interaction, in a cigar trap. The time-splitting and sine spectral method in space is proposed to discretize the time-dependent equations for computing the dynamics of dipolar BEC. The singularity in the dipole-dipole interaction brings significant difficulties both in mathematical analysis and in numerical simulations. Numerical results are given to show the efficiency of this method.

An Efficient Compact Finite Difference Method for the Solution of the Gross-Pitaevskii Equation

“…In recent years, a great deal of approaches have been proposed for computing the ground state. In fact, these approaches were divided into two types: one method is either based on solving the nonlinear eigenvalue problem [4]; the other approach is the imaginary time method which can be written as a normalized gradient flow formulation [5]. Choose the time step Δ > 0 and set = Δ , = 0, 1, 2, .…”

“…There are many methods discretizing the normalized gradient flow in the imaginary time. These methods include spectral (pseudospectral) methods [3,5], finite difference method (FDM) [6,7], and discontinuous Galerkin (DG) method [8][9][10].…”

N-Soliton Solutions of the Nonisospectral Generalized Sawada-Kotera Equation

“…Deng et al [6] and Sun et al [7,8] develop a systematic procedure to find soliton solutions of the nonisospectral equations. Based on exact solutions, numerical methods can be presented well for the nonisospectral nonlinear problem [9][10][11].…”