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標題: 利用延續演算法處理GPE方程之對稱性破壞解
Continuation algorithms for symmetry-breaking solutions of the Gross-Pitaevskii equation
作者: 林千資
Lin, Chien-Tzu
關鍵字: Bose-Einstein condensates;玻色愛因斯坦凝聚;semilinear elliptic eigenvalue problems;secondary bifurcations.;半線性橢圓特徵值問題;次分支解。
出版社: 應用數學系所
引用: [1] E. P. Gross, Hydrodynamics of a superfluid condensate, J. Math. Phys., 4 (1963) 195. [2] L. P. Pitaevskii, Vortex lines in an imperfect Bose gas, Zh. Eksp. Teor. Fiz., 13 (1961) 451. [3] R. D'Agosta and C. Presilla, States without a linear counterpart in Bose-Einstein condensates, Phys. Rev. A, 65 (2002) 043609. [4] J. C. Eilbeck, P. S. Lomdahl, and A. C. Scott, The discrete self-trapping equation, Physica. D, 16 (1985) 318. [5] G. L. Alfimov and D. A. Zezyulin, Monlinear modes for the Gross-Pitaevskii equation - a demonstrative computation approach, Nonlinearity, 20 (2007) 2075-2092. [6] L. Bauer, H. B. Keller, and E. L. Reiss, Multiple eigenvalues lead to secondary bifurcation, SIAM Rev., 17 (1975) 101-122. [7] S.-L. Chang, C.-S. Chien and B.-W. Jeng, Liapunov-Schmidt reduction and continuation for nonlinear Schr¨odinger equations, SIAM J. Sci. Comput., 29 (2007) 729-755. [8] S.-L. Chang, C.-S. Chien and B.-W. Jeng, Computing wave functions of nonlinear Schr¨odinger equations: a time-independent approach, J. Comput. Phys., 226 (2007) 104-130. [9] S.-L. Chang and C.-S. Chien, Adaptive continuation algorithms for computing energy levels of rotating Bose-Einstein condensates, Comput. Phys. Commun., 177 (2007) 707-719. [10] E. L. Allgower, K. B¨ohmer, and Z. Mei, A complete bifurcation scenario for the 2-D nonlinear Laplacian with Neumann boundary conditions on the unit square, in: Bifurcation and Chaos: Analysis, Algorithms, Applications, R. Seydel, F.W. Schneider, T. K¨upper, H. Troger (Eds.), Internat. Ser. Numer. Math. 97, 1-18, Birkh¨auser, Basel, 1991. [11] P. J. Budden and J. Norbury, A non-linear elliptic eigenvalue problem, J. Inst Math. Appl., 24 (1979) 9-33. [12] P. J. Budden and J. Norbury, Solution branches for nonlinear equilibrium problem-bifurcation and domain perturbation, IMA J. Appl. Math., 28 (1982) 109-129. [13] P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982) 441-467. [14] E. L. Allgower and C.-S. Chien, Continuation and local perturbation for multiple bifurcations, SIAM J. Sci. Stat. Comput., 7 (1986) 1265-1281. [15] C.-S. Chien, S.-Y. Gong and Z. Mei, Mode jumping in the von K´arm´an equations, SIAM J. Sci. Comput., 22 (2000) 1354-1385. [16] C.-S. Chien, Secondary bifurcations in the buckling problem, J. Comput. Appl. Math., 25 (1989) 277-287. [17] M. Golubitsky and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory, Springer-Verlag, Heidelberg, 1985. [18] E. Isaacson and H. B. Keller, Analysis of Numerical Methods, John Wiley & Sons, New York, 1966. [19] Z. Mei, Path following around corank-2 bifurcation points of a semilinear elliptic problems with symmetry, Computing, 47 (1991) 69-85. [20] H. B. Keller, Lectures on Numerical Methods in Bifurcation Problems, Springer-Verlag, Berlin, 1987. [21] G. H. Golub and C. F. Van Loan, Matrix Computations, The Johns Hopkins University Press, Baltimore, MD, 1996. [22] C.-S. Chien, Z.-Li. Weng and C.-L. Shen, Lanczos type methods for continuation problems, Numer. Linear Algebra Appl., 4 (1997) 23-41. [23] C. C. Paige and M. A. Saunders,A bidiagonalization algorithm for sparse linear equations and least squares problems, Report SOL 7819, Department of Operation Research, Standford University, Standford, California, 1975. [24] B. N. Parlett, A new look at the Lanczos algorithm for solving symmetric systems of linear equations, Linear Algebra Appl., 29, 323-346, 1980. [25] B. N. Parlett., The Symmemetric Eigenvalue Problems, Prentice-Hall, Englewood Cliffs, NJ,1980.
我們利用分支理論(Bifurcation Theory)探討Gross-Pitaevskii方程(GPE)的對稱性破壞解的數值解,在此我們考慮拋物線位能井和四個一組的位能井。我們也探討在光學晶格上的玻色—愛因斯坦凝態,藉由正弦函數或餘弦函數來描繪週期性位能。我們發展延續演算法,來追蹤離散化的GPE的第一激發態矩形解的曲線。我們利用係數矩陣的奇異性,沿著主要解的解曲線上偵測次分支解。在各種不同類型的二維GPE,我們的數值結果顯示對稱性破壞解是存在的。

We study symmetry-breaking solutions of the Gross-Pitaevskii equation (GPE) numerically from the viewpoint of bifurcation theory. Both the parabolic trapping potential and the guadruple-well trapping potential are considered. We also study Bose-Einstein condensates (BEC) in optical lattices, where the periodic potential described by the sine function or the cosine function is imposed on the GPE. We develop continuation algorithms for tracing the rectangular solution curves and the triangular solution curves of the first excited-state solutions of the discrete GPE.
In particular, we detect secondary bifurcations along the primary solution curves by monitoring the singularity of the coefficient matrix. Our numerical results show that symmetry-breaking solutions do exist for various types of the 2 dimension GPE mentioned above.
其他識別: U0005-1905200918300100
Appears in Collections:應用數學系所

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