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arXiv:2204.09189 · Journal of Applied Science and Engineering A

Numerical solution to the time-independent gross-Pitaevskii Equation

Tsogbayar Tsednee, B. Tsednee, Tsookhuu Khinayat

Published Dec 1, 2021• 1 citation
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Abstract

We solve the time-independent Gross-Pitaevskii equation modeling the Bose-Einstein condensate trapped in an anistropic harmonic potential using a pseudospectral method. Numerically obtained values for an energy and a chemical potential for the condensate with positive and negative scattering length have been compared with those from the literature. The results show that they are in good agreement when an atomic interaction is not too strong.

DOI
10.5564/jasea.v2i1.3493

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We solve the time-independent Gross-Pitaevskii equation modeling the Bose-Einstein condensate trapped in an anistropic harmonic potential using a pseudospectral method. Numerically obtained values for an energy and a chemical potential for the condensate with positive and negative scattering length have been compared with those from the literature. The results show that they are in good agreement when an atomic interaction is not too strong.
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Both the homotopy analysis method and Galerkin spectral method are applied to find the analytical solutions of the two-dimensional and time-independent Gross-Pitaevskii equation, a nonlinear Schrödinger equation used in describing the system of Bose-Einstein condensates trapped in a harmonic potential. The approximate analytical solutions are obtained successfully. Comparisons between the analytical solutions and the numerical solutions have been made. The results indicate that they are agreement very well with each other when the atomic interaction is not too strong.
We try to find the analytical solutions to the time-independent Gross-Pitaevskii equation, a nonlinear Schrödinger equation used in the simulation of Bose—Einstein condensates trapped in a harmonic potential. Both the homotopy analysis method and the Galerkin spectral method are applied. We investigate the one-dimensional case and obtain the approximate analytical solutions successfully. Comparison between the analytical solutions and the numerical solutions has been made. The results indicate that they agree very well with each other when the atomic interaction is not too strong.
We study certain stationary and time-evolution problems of trapped Bose-Einstein condensates using the numerical solution of the Gross-Pitaevskii (GP) equation with both spherical and axial symmetries. We consider time-evolution problems initiated by suddenly changing the interatomic scattering length or harmonic trapping potential in a stationary condensate. These changes introduce oscillations in the condensate which are studied in detail. We use a time iterative split-step method for the solution of the time-dependent GP equation, where all nonlinear and linear non-derivative terms are treated separately from the time propagation with the kinetic energy terms. Even for an arbitrarily strong nonlinear term this leads to extremely accurate and stable results after millions of time iterations of the original equation.
In this work we employ the split-step technique combined with a Legendre pseudospectral representation to solve various time-dependent GrossPitaevskii equations (GPE). Our findings based on the numerical accuracy of this approach applied for one-dimensional (1D) and two-dimensional (2D) problems show that it can provide accurate and stable solutions. Moreover, this approach has been applied to study the dynamics of the Bose-Einstein condensate which is modeled with the GPE. The breathing of condensate with a repulsive and attractive interactions trapped in 1D and 2D harmonic potentials has been simulated as well.