### Abstract

We develop a continuation block successive over-relaxation (BSOR)-Lanczos-Galerkin method for the computation of positive bound states of time-independent, coupled Gross-Pitaevskii equations (CGPEs) which describe a multi-component Bose-Einstein condensate (BEC). A discretization of the CGPEs leads to a nonlinear algebraic eigenvalue problem (NAEP). The solution curve with respect to some parameter of the NAEP is then followed by the proposed method. For a single-component BEC, we prove that there exists a unique global minimizer (the ground state) which is represented by an ordinary differential equation with the initial value. For a multi-component BEC, we prove that m identical ground/bound states will bifurcate into m different ground/bound states at a finite repulsive inter-component scattering length. Numerical results show that various positive bound states of a two/three-component BEC are solved efficiently and reliably by the continuation BSOR-Lanczos-Galerkin method.

Original language | English |
---|---|

Pages (from-to) | 439-458 |

Number of pages | 20 |

Journal | Journal of Computational Physics |

Volume | 210 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2005 Dec 10 |

### Fingerprint

### Keywords

- Continuation BSOR-Lanczos-Galerkin method
- Gauss-Seidel-type iteration
- Gross-Pitaevskii equation
- Multi-component Bose-Einstein condensate
- Nonlinear Schrödinger equation

### ASJC Scopus subject areas

- Numerical Analysis
- Modelling and Simulation
- Physics and Astronomy (miscellaneous)
- Physics and Astronomy(all)
- Computer Science Applications
- Computational Mathematics
- Applied Mathematics

### Cite this

*Journal of Computational Physics*,

*210*(2), 439-458. https://doi.org/10.1016/j.jcp.2005.04.019

**A continuation BSOR-Lanczos-Galerkin method for positive bound states of a multi-component Bose-Einstein condensate.** / Chang, Shu Ming; Kuo, Yuen Cheng; Lin, Wen Wei; Shieh, Shih-Feng.

Research output: Contribution to journal › Article

*Journal of Computational Physics*, vol. 210, no. 2, pp. 439-458. https://doi.org/10.1016/j.jcp.2005.04.019

}

TY - JOUR

T1 - A continuation BSOR-Lanczos-Galerkin method for positive bound states of a multi-component Bose-Einstein condensate

AU - Chang, Shu Ming

AU - Kuo, Yuen Cheng

AU - Lin, Wen Wei

AU - Shieh, Shih-Feng

PY - 2005/12/10

Y1 - 2005/12/10

N2 - We develop a continuation block successive over-relaxation (BSOR)-Lanczos-Galerkin method for the computation of positive bound states of time-independent, coupled Gross-Pitaevskii equations (CGPEs) which describe a multi-component Bose-Einstein condensate (BEC). A discretization of the CGPEs leads to a nonlinear algebraic eigenvalue problem (NAEP). The solution curve with respect to some parameter of the NAEP is then followed by the proposed method. For a single-component BEC, we prove that there exists a unique global minimizer (the ground state) which is represented by an ordinary differential equation with the initial value. For a multi-component BEC, we prove that m identical ground/bound states will bifurcate into m different ground/bound states at a finite repulsive inter-component scattering length. Numerical results show that various positive bound states of a two/three-component BEC are solved efficiently and reliably by the continuation BSOR-Lanczos-Galerkin method.

AB - We develop a continuation block successive over-relaxation (BSOR)-Lanczos-Galerkin method for the computation of positive bound states of time-independent, coupled Gross-Pitaevskii equations (CGPEs) which describe a multi-component Bose-Einstein condensate (BEC). A discretization of the CGPEs leads to a nonlinear algebraic eigenvalue problem (NAEP). The solution curve with respect to some parameter of the NAEP is then followed by the proposed method. For a single-component BEC, we prove that there exists a unique global minimizer (the ground state) which is represented by an ordinary differential equation with the initial value. For a multi-component BEC, we prove that m identical ground/bound states will bifurcate into m different ground/bound states at a finite repulsive inter-component scattering length. Numerical results show that various positive bound states of a two/three-component BEC are solved efficiently and reliably by the continuation BSOR-Lanczos-Galerkin method.

KW - Continuation BSOR-Lanczos-Galerkin method

KW - Gauss-Seidel-type iteration

KW - Gross-Pitaevskii equation

KW - Multi-component Bose-Einstein condensate

KW - Nonlinear Schrödinger equation

UR - http://www.scopus.com/inward/record.url?scp=23844552570&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=23844552570&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2005.04.019

DO - 10.1016/j.jcp.2005.04.019

M3 - Article

AN - SCOPUS:23844552570

VL - 210

SP - 439

EP - 458

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 2

ER -