Efficient spectral collocation and continuation methods for symmetry-breaking solutions of rotating Bose–Einstein condensates

Abstract

We describe an efficient spectral collocation method (SCM) for symmetry-breaking solutions of rotating Bose–Einstein condensates (BECs) which is governed by the Gross–Pitaevskii equation (GPE). The Lagrange interpolants using the Legendre–Gauss–Lobatto points are used as the basis functions for the trial function space. Some formulas for the derivatives of the basis functions are given so that the GPE can be efficiently computed. The SCMs are incorporated in the context of a predictor–corrector continuation algorithm for tracing primary and secondary solution branches of the GPE. Symmetry-breaking solutions are numerically presented for both rotating BECs, BECs in optical lattices, and two-component BECs in optical lattices. Our numerical results show that the numerical algorithm we propose in this paper outperforms the classical orthogonal Legendre polynomials.

Keywords:

• Gross–Pitaevskii equation
• first excited-state solution
• Legendre polynomials
• Lagrange interpolation
• two-component BECs

2010 AMS Subject Classifications:

• 65N35
• 35P30
• 35Q55

Acknowledgements

The authors are grateful to Prof. H.T. Huang for his valuable comments and suggestions on Section 2. We would like to thank two reviewers for their comments that have improved the original version of this paper. C.-S. Chien was supported by the National Science Council of ROC (Taiwan) through Project NSC 98-2115-M-231-001-MY3.