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標題: 利用擬譜方法解Gross-Pitaevskii方程數值解
Spectral collocation methods for the numerical solutions of Gross-Pitaevskii equation
作者: 張書豪
Chang, Shu-Hao
關鍵字: 擬譜方法;Legendre polynomials;Gross-Pitaevskii equation;Legendre-Gauss integration;Bose-Einstein condensation
出版社: 應用數學系所
引用: References [1] D. Gottlieb and S. A. Orszang, Numerical Analysis of Spectral Methods: Theory and Applications, SIAM, Philadelphia, 1977. [2] C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer-Verlag, New York, 1987. [3] L. N. Trefethen, Spectral Methods in Matlab, SIAM, 2000. [4] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman and E. A. Cornell, Observation of Bose-Einstein Condensation in a dilute atomic vapor, Science, 269 (1995), pp.198-201. [5] K. B. Davis, M. -O. Mewes, M. R. Andrews, N. J. Van Druten, D. S. Durfee, D. M. Kurn and W. Ketterle, Bose-Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett., 75 (1995), pp. 3969-3973. [6] S. Burger, F. S. Cataliotti, C. Fort, F. Minardi, M. Inguscio, M. L. Chiofalo and M. P. Tosi, Superfluid and dissipative dynamics of a Bose-Einstein condensate in a periodic potential, Phys. Rev. Lett., 86 (2001), pp. 4447-4450. [7] M. Cristiani, O. Morsch, J. H. Muller, D. Ciampini and E. Arimondo, Experimental properties of Bose-Einstein condensates in one-dimensional optical lattices: Bloch oscillations, Landau-Zener tunneling, and mean-filed effects, Phys. Rev. A, 65 (2002), pp. 063612 (11). [8] L. Fallani, F. S. Cataliotti, J. Catani, C. Fort, M. Modugno, M. Zawada and M. Inguscio, Optically induced lensing effect on a Bose-Einstein condensate expanding in a moving lattice, Phys. Rev. Lett., 91 (2003), pp. 24045 (4). [9] B. -W. Jeng and C. -S. Chien, Spectral continuation methods using Chebyshev polynomials for nonlinear Schr¨odinger equations, submitted to SIAM J. Sci. Comput, 2010. [10] R. L. Burden and J. D. Faires, Numerical analysis, (7th ed.), 2001. [11] H.-S. Chen and C.-S. Chien, Multi-level spectral-Galerkin and continuation methods for nonlinear Schr¨odinger equations, Multiscale Model. Simul., 8 (2009), pp. 370-392.
爲了求Gross-Pitaevskii 方程式的數值解,我們描述一個有效的擬譜方法,其中Legendre多項式作爲試驗函數空間的基底函數。我們說明運用此種基底函數於擬譜方法和涉及到Gauss-Legendre積分的Galerkin–譜方法會等價。我們結合擬譜方法於延續法之中來計算定態非線性薛丁格方程的能階和波函數。我們亦對玻色–愛因斯坦凝聚態在週期位能下的數值結果提出報告。

We describe an efficient spectral collocation method (SCM) for the numerical solutions of the Gross-Pitaevskii equation (GPE), where the Legendre polynomials are used as the basis functions for the trial function space. We show that the SCM obtained in this way is equivalent to the spectral- Galerkin method (SGM) involving the Gauss-Legendre integration. The SCM is incorporated in the context of continuaton methods for computing energy levels and wave functions of the stationary state nonlinear Schr¨odinger equation. Numerical results on the Bose-Einstein condensates (BEC) in a periodic potential are reported.
其他識別: U0005-2407201016232100
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