Computing the Ground State Solution of Bose--Einstein Condensates by a Normalized Gradient Flow

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.

Universally‐Convergent Squared‐Operator Iteration Methods for Solitary Waves in General Nonlinear Wave Equations

“…µ n+1 = µ n + u, M −2 L 0 u − α n θ n H n u=un, µ=µn ∆t, (A.15) where 17) and (G n , H n ) are user specified functions such as (10).…”

“…If one views the eigenvalues as propagation constants, and requires the eigenfunctions to have a fixed power(norm), then this linear eigenvalue problem becomes the same as a solitary-wave problem with a pre-specified power. To treat this type of problems, one can use the imaginary-time evolution method [9,10,11,12,13], where the solution is normalized at every step to keep the pre-specified power. However, the problem with the imaginary-time evolution method is that it often diverges when the solution crosses zero [13].…”

“…Combining ideas of SOM (4) and the imaginary-time evolution method [9,10,13], we propose the following new power-conserving squared-operator method (PCSOM):…”

“…[9,10,12], where imaginary-time evolution methods for vector equations were reported. Of course, the corresponding imaginary-time evolution methods, unlike the PCSOM (34)-(38) and QCSOM (49), will not be universally-convergent.…”

“…This method can become cumbersome as well in high dimensions when the linear eigenvalue problem becomes harder to solve. Two more methods are the Petviashvili method [5,6,7,8] and the imaginary-time evolution method [9,10,11,12,13]. The convergence properties of the former method were studied in [6] for a class of equations with power nonlinearity, while those of the latter method were obtained in [13] for a much larger class of equations with arbitrary nonlinearity.…”