| Summary: | 博士 === 國立中興大學 === 應用數學系所 === 100 === In this dissertation, we study various types of numerical continuation methods for computing numerical solutions of multi-parameter nonlinear eigenvalue problems. These include two-grid continuation algorithms, spectral Galerkin and continuation algorithms, and spectral collocation and continuation algorithms. For the last two types of continuation algorithms, we use the second kind Chebyshev polynomials as the basis functions for the trial function space. The proposed algorithms are exploited to compute symmetry-breaking solutions of the Gross-Pitaevskii equation, the ground state solutions of rotating Bose-Einstein condensates (BEC), and rotating two-component BECs.
First, We present an efficient spectral-Galerkin continuation method (SGCM) and two-grid centered difference approximations for symmetry-breaking solutions of the GPE. Some basic formulae for the SGCM are derived so that the eigenvalues of the associated linear eigenvalue problems can be easily computed. The SGCM is implemented to investigate the ground and first excited state solutions of the GPE. Both the parabolic and quadruple-well trapping potentials are considered. We also study BEC in optical lattices, where the periodic potential described by the sine or cosine functions is imposed on the GPE.
Next, we study spectral collocation continuation method (SCCM) for rotating BEC and rotating BEC in optical lattices. The ground state and first excited state solutions of rotating BEC in optical lattices are investigated.
Finally, We study efficient SCCM for rotating two-component BECs and rotating two-component BECs in optical lattices. A novel two-parameter continuation algorithm is proposed for computing the ground state and first excited state solutions of the governing GPEs, where the classical tangent vector is split into two constraint conditions for the bordered linear systems. Numerical results on rotating two-component BECs and rotating two-component BECs in optical lattices are reported. The results on the former are consistent with published numerical results.
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