Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Chinese Physics B

Application of the homotopy analysis method for the Gross—Pitaevskii equation with a harmonic trap

Yu-Ren Shi, Cong-Bo Liu, Guanghui Wang, Zhigang Zhou

Published Dec 1, 2012• 3 citations
Publisher

Abstract

The Homotopy analysis method (HAM) is adopted to find the approximate analytical solutions of the Gross—Pitaevskii equation, a nonlinear Schrödinger equation is used in simulation of Bose—Einstein condensates trapped in a harmonic potential. Comparisons between the analytical solutions and the numerical solutions have been made. The results indicate that they fit very well with each other when the atomic interaction is weak.

DOI
10.1088/1674-1056/21/12/120307

Related papers

Analytical Solutions to the Time-Independent Gross-Pitaevskii Equation with a Harmonic Trap

Yu-Ren Shi, Guanghui Wang, Cong-Bo Liu +2 more

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.

Chinese Physics Letters•Nov 1, 2012•3 citations

Analytical Solutions for the Two-Dimensional Gross—Pitaevskii Equation with a Harmonic Trap

Yu-Ren Shi, Xue-Ling Wang, Guanghui Wang +2 more

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.

Communications in Theoretical Physics•Mar 1, 2013•4 citations

Analytical solutions for the spin-1 Bose-Einstein condensate in a harmonic trap

Yu-Ren Shi, Xue-Ling Wang, Guanghui Wang +3 more

The homotopy analysis method and Galerkin spectral method are applied to find the analytical solutions for the Gross-Pitaevskii equations, a set of nonlinear Schrödinger equation used in simulation of spin-1 Bose-Einstein condensates trapped in a harmonic potential. We investigate the one-dimensional case and get the approximate analytical solutions successfully. Comparisons between the analytical solutions and the numerical solutions have been made. The results indicate that they are in agreement well with each other when the atomic interaction is weakly. We also find a class of exact solutions for the stationary states of the spin-1 system with harmonic potential for a special case.

Frontiers of Physics•May 4, 2013•2 citations

Numerical Solution of Two-dimensional Gross-Pitaevskii Equation for Bose-Einstein Condensation in a Non-harmonic Trap

Zhao Jiangang

A two-dimensional Gross-Pitaevskii equation describing Bose-Einstein condensation in an axial symmetry non-harmonic trap is solved.Converged wave functions,chemical potentials,and internal energies in Bose-Einstein condensation are obtained.Ground-state of Bose-Einstein condensation with atomic interaction is investigated.A comparison between harmonic and non-harmonic traps is reported.

Jisuan wuli•Jan 1, 2009•0 citations

Spectral method for the time-dependent Gross-Pitaevskii equation with a harmonic trap

Claude M. Dion, Eric Cancès B

We study the numerical resolution of the time-dependent Gross-Pitaevskii equation, a nonlinear Schrödinger equation used to simulate the dynamics of Bose-Einstein condensates. Considering condensates trapped in harmonic potentials, we present an efficient algorithm by making use of a spectral-Galerkin method, using a basis set of harmonic-oscillator functions, and the Gauss-Hermite quadrature. We apply this algorithm to the simulation of condensate breathing and scissor modes.

arXiv:cond-mat/0209646•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Apr 15, 2003•57 citations

The application of the “inverse problem” method for constructing confining potentials that make N-soliton waveforms exact solutions in the Gross–Pitaevskii equation

Fred Cooper, Avinash Khare, John F. Dawson +2 more

In this work, we discuss an application of the "inverse problem" method to find the external trapping potential, which has particular N trapped soliton-like solutions of the Gross-Pitaevskii equation (GPE) also known as the cubic nonlinear Schrödinger equation (NLSE). This inverse method assumes particular forms for the trapped soliton wave function, which then determines the (unique) external (confining) potential. The latter renders these assumed waveforms exact solutions of the GPE (NLSE) for both attractive (g<0) and repulsive (g>0) self-interactions. For both signs of g, we discuss the stability with respect to self-similar deformations and translations. For g<0, a critical mass Mc or equivalently the number of particles for instabilities to arise can often be found analytically. On the other hand, for the case with g>0 corresponding to repulsive self-interactions which is often discussed in the atomic physics realm of Bose-Einstein condensates, the bound solutions are found to be always stable. For g<0, we also determine the critical mass numerically by using linear stability or Bogoliubov-de Gennes analysis, and compare these results with our analytic estimates. Various analytic forms for the trapped N-soliton solutions in one, two, and three spatial dimensions are discussed, including sums of Gaussians or higher-order eigenfunctions of the harmonic oscillator Hamiltonian.

Chaos An Interdisciplinary Journal of Nonlinear Science•Apr 1, 2024•2 citations
The Homotopy analysis method (HAM) is adopted to find the approximate analytical solutions of the Gross—Pitaevskii equation, a nonlinear Schrödinger equation is used in simulation of Bose—Einstein condensates trapped in a harmonic potential. Comparisons between the analytical solutions and the numerical solutions have been made. The results indicate that they fit very well with each other when the atomic interaction is weak.
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.
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.
The homotopy analysis method and Galerkin spectral method are applied to find the analytical solutions for the Gross-Pitaevskii equations, a set of nonlinear Schrödinger equation used in simulation of spin-1 Bose-Einstein condensates trapped in a harmonic potential. We investigate the one-dimensional case and get the approximate analytical solutions successfully. Comparisons between the analytical solutions and the numerical solutions have been made. The results indicate that they are in agreement well with each other when the atomic interaction is weakly. We also find a class of exact solutions for the stationary states of the spin-1 system with harmonic potential for a special case.
In this work, we discuss an application of the "inverse problem" method to find the external trapping potential, which has particular N trapped soliton-like solutions of the Gross-Pitaevskii equation (GPE) also known as the cubic nonlinear Schrödinger equation (NLSE). This inverse method assumes particular forms for the trapped soliton wave function, which then determines the (unique) external (confining) potential. The latter renders these assumed waveforms exact solutions of the GPE (NLSE) for both attractive (g<0) and repulsive (g>0) self-interactions. For both signs of g, we discuss the stability with respect to self-similar deformations and translations. For g<0, a critical mass Mc or equivalently the number of particles for instabilities to arise can often be found analytically. On the other hand, for the case with g>0 corresponding to repulsive self-interactions which is often discussed in the atomic physics realm of Bose-Einstein condensates, the bound solutions are found to be always stable. For g<0, we also determine the critical mass numerically by using linear stability or Bogoliubov-de Gennes analysis, and compare these results with our analytic estimates. Various analytic forms for the trapped N-soliton solutions in one, two, and three spatial dimensions are discussed, including sums of Gaussians or higher-order eigenfunctions of the harmonic oscillator Hamiltonian.