Please use this identifier to cite or link to this item:
標題: 利用階梯式多重網格延續法處理具週期位能的Gross-Pitaevskii方程
A staircase multigrid-continuation algorithm for computing energy levels of the Gross-Pitaevskii equation in a periodic potential
作者: 王信舜
Wang, Hsin-Shun
關鍵字: nonlinear Schr&ouml
dinger equation
two-grid scheme
continuation method
出版社: 應用數學系所
引用: [1] 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, 198-201 (1995). [2] K. B. Davis, M. -O. Mewes, M. R. Anderws, N. J. Van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Bose-Einstein condensation in a sodium atoms, Phys. Rev. Lett., 75, 3969-3973 (1995). [3] D. A. Butts and D. S. Rokhsar, Predicted signatures of rotating Bose-Einstein condensations, Nature, 397, 327-329 (1999). [4] M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S, Hall, C. E. Wieman, and E. A. Cornell, Vortices in a Bose-Einstein condensate, Phys. Rev. Lett., 83, 2498-2501 (1999). [5] L. Deng, E. W. Hagley, J. Wen, M. Trippenbach, Y. Band, F. E. Simsarian, K. Helmerson, S. L. Rolston, and W. D. Philips, Four-wave mixing with matter waves, Nature, 398, 218-220 (1999). [6] E. P. Gross, Structure of a quantized vortex in boson systems, Nuovo. Cimento., 20, 454-477 (1961). [7] L. P. Pitaevskii, Vortex lines in an imperfect bose gas, Soviet Phys. JETP, 13, 451-454 (1961). [8] W. Bao and Q. Du, Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput., 25, 1674-1697 (2004). [9] J. J. José García-Ripoll and V. M. Pérez-García, Optimizing Schrödinger functionals using Sobolev gradients: Applications to guantum mechanics and nonlinear optics, SIAM J. Sci. Comput., 23, 1316-1334 (2001). [10] W. Bao and F. Y. Lim, Computing ground states of spin-1 Bose-Einstein condensates by the normalized gradient flow, SIAM J. Sci Comput., 30, No. 4, 1925-1948 (2008). [11] J. Yang and T. I. Lakoba, Accelerated imaginary-time evolution methods for the computation of solitary waves, Stud. Appl. Math., 120, 265-292 (2008). [12] T. I. Lakoba and J. Yang, A generalized Petviashvili iteration method for scalar and vector Hamiltonian equations with arbitrary form of nonlinearity, J. Comput. Phys., 226, 1668-1692 (2007). [13] V. I. Petviashvili, Equation of an extraordinary soliton, Plasma Phys., 2, 469-472 (1976). [14] T. Kapitula and P. G. Kevrekidis, Bose-Einstein condensates in the presence of a magnetic trap and optical lattice, Chaos, 15, 037114 (2005). [15] J. C. Bronski, L. D. Carr, R. Carretero-González, B. Deconinck, J. N. Kutz, and K. Promislow, Stability of attractive Bose-Einstein condensates in a periodic potential, Phys. Rev. E, 64, 056615 (2001). [16] C. -S. Chien and B. -W. Jeng, A two-grid discretization scheme for semilinear elliptic eigenvalue problems, SIAM J. Sci. Comput., 27, 1287-1304 (2006). [17] C. -S. Chien, H. -T. Huang, B. -W. Jeng, and Z. -C. Li, Two-grid discretization schemes for nonlinear Schrödinger equations, J. Comput. Appl. Math., 214, 549-571 (2008). [18] S. -L. Chang, C. -S. Chien, and B. -W. Jeng, Liapunov-Schmidt reduction and continuation for nonlinear Schrödinger equations, SIAM J. Sci. Comput., 29, 729-755 (2007). [19] S. -L. Chang, C. -S. Chien, and B. -W. Jeng, Computing wave functions of nonlinear Schrödinger equations: a time-independent approach, J. Comput. Phys., 226, 104-130 (2007). [20] S. -L. Chang, C. -S. Chien, and Z. -C. Li, A finite difference continuation method for computing energy levels of Bose-Einstein condensates, Comput. Phys. Commun., 179, 208-226 (2008). [21] S. Burger, F. S. Cataliotti, C. Fort, F. Minardi, and M. Inguscio, Superfluid and dissipative dynamics of a Bose-Einstein condensate in a periodic potential, Phys. Rev. Lett., 86, 4447-4450 (2001). [22] 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, 063612 (10) (2001). [23] 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, 240450 (4) (2003). [24] C. Gui and J. Wei, Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Diff. Eqs., 158, 1-27 (1999). [25] M. del Pino, M. Kowalczyk, and J. Wei, Multi-bump ground states of the Gierer-Meinhardt systems in R^{2}, Ann. I. H. Poincaré, AN 20, 53-85 (2003). [26] G. L. G. Sleijpen and H. A. van der Vorst, A Jacobi-Davidson iteration method for linear eigenvalue problems, SIAM J. Matrix Anal. Appl., 17, 401-425 (1996). [27] G. L. G. Sleijpen, A. G. L. Booten, D. R. Fokkema, and H. A. van der Vorst, Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems, BIT, 36, 595-633 (1996). [28] C. -S. Chien, B. -W. Jeng, and Z. -C. Li, A time-independent approach for computing wave functions of the Schrödinger-Poisson system, Numer. Linear Algebra Appl., 15, 55-82 (2008). [29] S. -L. Chang, C. -S. Chien, and B. -W. Jeng, An efficient algorithm for the Schrödinger-Poisson eigenvalue problem, J. Comput. Appl. Math., 205, 509-532 (2007). [30] S. Costiner and S. Ta'asan, Adaptive multigrid techniques for large-scale eigenvalue problems: solutions of the Schrödinger problem in two and three dimensions, Phys. Rev. E, 51, 3704-3717 (1995). [31] S. Costiner and S. Ta'asan, Simultaneous multigrid techniques for nonlinear eigenvalue problems: solutions of the nonlinear Schrödinger-Poisson eigenvalue problem in two and three dimensions, Phys. Rev. E, 52, 1181-1192 (1995). [32] S. -L. Chang and C. -S. Chien, Adaptive continuation algorithms for computing energy levels of rotating Bose-Einstein condensates, Computer Phys. Commun., 177, 707-719 (2007). [33] R. D'Agosta, B. A. Malomed, and C. Presilla, Stationary states of Bose-Einstein condensates in single- and multi-well trapping potentials, Laser Phys., 12, 37-42 (2002). [34] R. D'Agosta and C. Presilla, States without a linear counterpart in Bose-Einstein condensates, Phys. Rev. A, 65, 043609 (2002). [35] G. L. Alfimov and D. A. Zezyulin, Nonlinear modes for the Gross-Pitaevskii equation-a demonstrative computation approach, Nonlinearity, 20, 2075-2092 (2007). [36] E. L. Allgower and C. -S. Chien, Continuation and local perturbation for multiple bifurcations, SIAM J. Sci. Stat. Comput., 7, 1265-1281 (1986). [37] C. -S. Chien, S. -Y. Gong, and Z. Mei, Mode jumping in the von Kármán equations, SIAM J. Sci. Comput., 22, 1354-1385 (2000).
摘要: 我們利用階梯式多重網格延續法處理具週期位能的玻色–愛因斯坦凝體的能階問題。此演算法是為了追蹤描繪半線性橢圓特徵值問題的解曲線,它是雙重網格離散法或是簡化雙重網格法的改進。此演算法與(簡化)雙重網格法比較之下有下列優點:(i)保證在最細的網格上收斂到目標點。(ii)它比簡化雙重網格法便宜。我們應用此演算法去計算一維和二維擁有週期位能的玻色–愛因斯坦凝體的基態解、第一激發態解以及對稱性破壞解。我們的數值結果驗證在定義域Ω=(-l,l)^{n},且化學位能夠大時,則其基態解的尖峰個數為Π_{j=1}^{n}(2l/d_{j}-1),其中d_{1}是在x座標上鄰近的位能井的距離,以此類推。此外,我們研究光子晶體的禁制帶光孤子,並報告數值結果。
We describe a staircase multigrid-continuation algorithm for computing energy levels of Bose-Einstein condensates (BEC) in a periodic potential. The proposed algorithm is a modification of the two-grid discretization schemes [16] or the simplified two-grid schemes [32] for tracing solution curves of semilinear elliptic eigenvalue problems. The algorithm has the following advantages over the (simplified) two-grid scheme: (i) It guarantees that the scheme will converge to the target point on the finest grid. (ii) It is cheaper than the simplified two-grid scheme. We apply the proposed algorithm to compute the ground state solutions, the first excited solutions, and symmetry-breaking solutions of the 1D BEC and 2D BEC in a periodic potential. Our numerical results show that if the domain Ω=(-l,l)^{n}, and the chemical potential is large enough, the number of peaks for the ground state solutions is Π_{j=1}^{n}(2l/d_{j}-1), where d_{1} is the distance of neighbor wells in the x-coordinate, and so on, and n the dimension of the BEC. Additionally, we study gap solitons in nonlinear photonic crystals. Some numerical results are reported.
其他識別: U0005-1905200916515900
Appears in Collections:應用數學系所



Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.