| Summary: | 博士 === 國立中興大學 === 應用數學系所 === 102 === In this dissertation, we study numerical solutions of Bose-Einstein condensates (BECs) with different physical phenomenon, which are governed by various types of the Gross-Pitaevskii equations (GPEs).
First, we describe the spectral collocation methods (SCMs) for numerical solutions of the GPE. The Lagrange interpolants using the Legendre-Gauss-Lobatto points are used for the basis functions. We give some formulae for the derivatives of the Lagrange interpolants for the Laplacian. Thus the linear term of the nonlinear Schr‥odinger equation (NLS) can be easily evaluated. We exploited the SCM to investigate the ground state and first excited state solutions for a rotating BEC and a rotating BEC in optical lattices.
Next, we describe an efficient SCM to compute symmetry-breaking solutions of the first excited state solutions of rotating BEC, rotating BEC in optical lattices, two-component BECs and two-component BECs in optical lattices.
Finally, we present multi-parameter/multiscale continuation methods combined with SCM for computing numerical solutions of rotating two-component BECs, which are governed by the GPEs. Various types of orthogonal polynomials are used as the basis functions for the trial function space. A novel multi-parameter/multiscale continuation algorithm is proposed for computing the solutions of the governing GPEs, where the chemical potential of each component and angular velocity are treated as the continuation parameters simultaneously. The proposed algorithm can effectively compute numerical solutions with abundant physical phenomena.
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