Mathematical theory and numerical methods for Bose-Einstein condensation

Kinetic and Related Models

Volumn 6, Issue 1, 2013, Pages 1-135

  Bao, Weizhu ^a   Cai, Yongyong ^a,b  

^a NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

^b Research Center   (China)

Author keywords

Bose Einstein condensation; Dynamics; Error estimate; Gross Pitaevskii equation; Ground state; Numerical method; Quantized vortex

Indexed keywords

EID: 84872170647     PISSN: 19375093     EISSN: 19375077     Source Type: Journal    
DOI: 10.3934/krm.2013.6.1     Document Type: Article

References (198)

1. J. R. Abo-Shaeer, C. Raman, J. M. Vogels and W. Ketterle, Observation of vortex lattices in Bose-Einstein condensates, Science, 292 (2001), 476-479.
2. S. K. Adhikari and P. Muruganandam, Bose-Einstein condensation dynamics from the numerical solution of the Gross-Pitaevskii equation, J. Phys. B: At. Mol. Opt. Phys., 35 (2002), 2831-2843.
3. S. K. Adhikari and P. Muruganandam, Mean-field model for the interference of matter-waves from a three-dimensional optical trap, Phys. Lett. A, 310 (2003), 229-235.
4. A. Aftalion, "Vortices in Bose-Einstein Condensates," Progress in Nonlinear Differential Equations and their Applications, 67, Birkhäuser, Boston, 2006.
5. A. Aftalion and Q. Du, Vortices in a rotating Bose-Einstein condensate: Critical angular velocities and energy diagrams in the Thomas-Fermi regime, Phys. Rev. A, 64 (2001), 063603.
6. A. Aftalion, Q. Du and Y. Pomeau, Dissipative flow and vortex shedding in the Painlevé boundary layer of a Bose Einstein condensate, Phys. Rev. Lett., 91 (2003), 090407.
7. A. Aftalion, R. Jerrard and J. Royo-Letelier, Non-existence of vortices in the small density region of a condensate, J. Funct. Anal., 260 (2011), 2387-2406.
8. K. Aikawa, A. Frisch, M. Mark, S. Baier, A. Rietzler, R. Grimm and F. Ferlaino, Bose-Einstein condensation of Erbium, Phys. Rev. Lett., 108 (2012), 210401.
9. G. Akrivis, Finite difference discretization of the cubic Schrödinger equation, IMA J. Numer. Anal., 13 (1993), 115-124.
10. G. Akrivis, V. Dougalis and O. Karakashian, On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation, Numer. Math., 59 (1991), 31-53.
11. J. O. Andersen, Theory of the weakly interacting Bose gas, Rev. Mod. Phys., 76 (2004), 599-639.
12. M. H. Anderson, J. R. Ensher, M. R. Matthewa, C. E. Wieman and E. A. Cornell, Observation of Bose-Einstein condensation in a dilute atomic vapor, Science, 269 (1995), 198-201.
13. P. Antonelli, D. Marahrens and C. Sparber, On the Cauchy problem for nonlinear Schrödinger equations with rotation, Discrete Contin. Dyn. Syst., 32 (2012), 703-715.
14. W. Bao, The nonlinear Schr̈odinger equation and applications in Bose-Einstein condensation and plasma physics, in "Dynamics in Models of Coarsening, Coagulation, Condensation and Quantization" (IMS Lecture Notes Series, World Scientific), 9 (2007), 141-240.
15. W. Bao, Ground states and dynamics of multicomponent Bose-Einstein condensates, Multiscale Model. Simul., 2 (2004), 210-236.
16. W. Bao, Numerical methods for the nonlinear Schrödinger equation with nonzero far-field conditions, Methods Appl. Anal., 11 (2004), 367-387.
17. W. Bao, Analysis and efficient computation for the dynamics of two-component Bose-Einstein condensates: Stationary and time dependent Gross-Pitaevskii equations, Contemp. Math., 473 (2008), 1-26.
18. W. Bao, N. Ben Abdallah and Y. Cai, Gross-Pitaevskii-Poisson equations for dipolar Bose-Einstein condensate with anisotropic confinement, SIAM J. Math. Anal., 44 (2012), 1713-1741.
19. W. Bao and Y. Cai, Ground states of two-component Bose-Einstein condensates with an internal atomic Josephson junction, East Asia J. Appl. Math., 1 (2010), 49-81.
20. W. Bao and Y. Cai, Uniform error estimates of finite difference methods for the nonlinear Schrödinger equation with wave operator, SIAM J Numer. Anal., 50 (2012), 492-521.
21. W. Bao and Y. Cai, Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation, Math. Comp., 82 (2013), 99-128.
22. W. Bao and Y. Cai, Uniform and optimal error estimates of an exponential wave integrator sine pseudospectral method for the nonlinear Schrödinger equation with wave operator, preprint.
23. W. Bao, Y. Cai and H. Wang, Efficient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates, J. Comput. Phys., 229 (2010), 7874-7892.
24. W. Bao and M. H. Chai, A uniformly convergent numerical method for singularly perturbed nonlinear eigenvalue problems, Commun. Comput. Phys., 4 (2008), 135-160.
25. W. Bao, I. Chern and F. Y. Lim, Efficient and spectrally accurate numerical methods for computing ground and first excited states in Bose-Einstein condensates, J. Comput. Phys., 219 (2006), 836-854.
26. W. Bao, I. Chern and Y. Zhang, Efficient methods for computing ground states of spin-1 Bose-Einstein condensates based on their characterizations, preprint.
27. W. Bao and Q. Du, Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput., 25 (2004), 1674-1697.
28. W. Bao, Q. Du and Y. Zhang, Dynamics of rotating Bose-Einstein condensates and its efficient and accurate numerical computation, SIAM J. Appl. Math., 66 (2006), 758-786.
29. W. Bao, Y. Ge, D. Jaksch, P. A. Markowich and R. M. Weishäupl, Convergence rate of dimension reduction in Bose-Einstein condensates, Comput. Phys. Comm., 177 (2007), 832-850.
30. W. Bao and D. Jaksch, An explicit unconditionally stable numerical method for solving damped nonlinear Schrödinger equations with a focusing nonlinearity, SIAM J. Numer. Anal., 41 (2003), 1406-1426.
31. W. Bao, D. Jaksch and P. A. Markowich, Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation, J. Comput. Phys., 187 (2003), 318-342.
32. W. Bao, D. Jaksch and P. A. Markowich, Three dimensional simulation of jet formation in collapsing condensates, J. Phys. B: At. Mol. Opt. Phys., 37 (2004), 329-343.
33. W. Bao, S. Jin and P. A. Markowich, On time-splitting spectral approximation for the Schrödinger equation in the semiclassical regime, J. Comput. Phys., 175 (2002), 487-524.
34. W. Bao, S. Jin and P. A. Markowich, Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimes, SIAM J. Sci. Comput., 25 (2003), 27-64.
35. W. Bao, H.-L. Li and J. Shen, A generalized Laguerre-Fourier-Hermite pseudospectral method for computing the dynamics of rotating Bose-Einstein condensates, SIAM J. Sci. Comput., 31 (2009), 3685-3711.
36. 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 (2008), 1925-1948.
37. W. Bao, F. Y. Lim and Y. Zhang, Energy and chemical potential asymptotics for the ground state of Bose-Einstein condensates in the semiclassical regime, Bull. Inst. Math. Acad. Sin. (N.S.), 2 (2007), 495-532.
38. W. Bao, P. A. Markowich, C. Schmeiser and R. M.Weishäupl, On the Gross-Pitaevskii equation with strongly anisotropic confinement: Formal asymptotics and numerical experiments, Math. Models Meth. Appl. Sci., 15 (2005), 767-782.
39. W. Bao, L. Pareschi and P. A. Markowich, Quantum kinetic theory: modeling and numerics for Bose-Einstein condensation, in "Modeling and Computational Methods for Kinetic Equations" (Birkhäuser Series: Modeling and Simulation in Science, Engineering and Technology), (2004), 287-321.
40. W. Bao and J. Shen, A fourth-order time-splitting Laguerre-Hermite pseudospectral method for Bose-Einstein condensates, SIAM J. Sci. Comput., 26 (2005), 2020-2028.
41. W. Bao and J. Shen, A generalized-Laguerre-Hermite pseudospectral method for computing symmetric and central vortex states in Bose-Einstein condensates, J. Comput. Phys., 227 (2008), 9778-9793.
42. W. Bao, Q. Tang and Z. Xu, Numerical methods and comparison for computing dark and bright solitons in the nonlinear Schrödinger equation, J. Comput. Phys., to appear.
43. W. Bao and W. Tang, Ground-state solution of Bose-Einstein condensate by directly minimizing the energy functional, J. Comput. Phys., 187 (2003), 230-254.
44. W. Bao and H. Wang, An efficient and spectrally accurate numerical method for computing dynamics of rotating Bose-Einstein condensates, J. Comput. Phys., 217 (2006), 612-626.
45. W. Bao and H. Wang, A mass and magnetization conservative and energy-diminishing numerical method for computing ground state of spin-1 Bose-Einstein condensates, SIAM J. Numer. Anal., 45 (2007), 2177-2200.
46. W. Bao, H. Wang and P. A. Markowich, Ground, symmetric and central vortex states in rotating Bose-Einstein condensates, Commun. Math. Sci., 3 (2005), 57-88.
47. W. Bao and Y. Zhang, Dynamics of the ground state and central vortex states in Bose-Einstein condensation, Math. Models Methods Appl. Sci., 15 (2005), 1863-1896.
48. W. Bao and Y. Zhang, Dynamical laws of the coupled Gross-Pitaevskii equations for spin-1 Bose-Einstein condensates, Methods Appl. Anal., 17 (2010), 49-80.
49. M. A. Baranov, M. Dalmonte, G. Pupillo and P. Zolle, Condensed matter theory of dipolar quantum gases, Chem. Rev., 112 (2012), 5012-5061.
50. M. D. Barrett, J. A. Sauer and M. S. Chapman, All-optical formation of an atomic Bose-Einstein condensate, Phys. Rev. Lett., 87 (2001), 010404.
51. N. Ben Abdallah, Y. Cai, F. Castella and F. Méhats, Second order averaging for the nonlinear Schrödinger equation with strongly anisotropic potential, Kinet. Relat. Models, 4 (2011), 831-856.
52. N. Ben Abdallah, F. Castella and F. Méhats, Time averaging for the strongly confined nonlinear Schrödinger equation, using almost periodicity, J. Diff. Eqn., 245 (2008), 154- 200.
53. N. Ben Abdallah, F. Méhats, C. Schmeiser and R. M. Weishäupl, The nonlinear Schrödinger equation with a strongly anisotropic harmonic potential, SIAM J. Math. Anal., 37 (2005), 189-199.
54. C. Besse, B. Bidégaray and S. Descombes, Order estimates in time of splitting methods for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 40 (2002), 26-40.
55. I. Bialynicki-Birula and Z. Bialynicki-Birula, Center-of-mass motion in the many-body theory of Bose-Einstein condensates, Phys. Rev. A, 65 (2002), 063606.
56. I. Bloch, J. Dalibard and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys., 80 (2008), 885-964.
57. N. N. Bogoliubov, On the theory of superfluidity, J. Phys. USSR, 11 (1947), 23-32.
58. S. N. Bose, Plancks gesetz und lichtquantenhypothese, Zeitschrift fr Physik, 3 (1924), 178- 181.
59. C. C. Bradley, C. A. Sackett, J. J. Tollett and R. G. Hulet, Evidence of Bose-Einstein condensation in an atomic gas with attractive interaction, Phys. Rev. Lett., 75 (1995), 1687-1690.
60. C. C. Bradley, C. A. Sackett and R. G. Hulet, Bose-Einstein condensation of Lithium: Observation of limited condensates, Phys. Rev. Lett., 78 (1997), 985-989.
61. J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz and K. Promislow, Stability of repulsive Bose-Einstein condensates in a periodic potential, Phys. Rev. E, 63 (2001), 036612.
62. M. Bruderer, W. Bao and D. Jaksch, Self-trapping of impurities in Bose-Einstein condensates: Strong attractive and repulsive coupling, EPL, 82 (2008), 30004.
63. L. Cafferelli and F. H. Lin, An optimal partition problem for eigenvalues, J. Sci. Comput., 31 (2007), 5-18.
64. Yongyong Cai, "Mathematical Theory and Numerical Methods for the Gross-Piatevskii Equations and Applications," Ph.D Thesis, National Universtiy of Singapore, 2011.
65. Y. Cai, M. Rosenkranz, Z. Lei and W. Bao, Mean-field regime of trapped dipolar Bose-Einstein condensates in one and two dimensions, Phys. Rev. A, 82 (2010), 043623.
66. M. Caliari and M. Squassina, Location and phase segregation of ground and excited states for 2D Gross-Pitaevskii systems, Dynamics of PDE, 5 (2008), 117-137.
67. P. Capuzzi and S. Hernandez, Bose-Einstein condensation in harmonic double wells, Phys. Rev. A, 59 (1999), 1488.
68. B. M. Caradoc-Davis, R. J. Ballagh and K. Burnett, Coherent dynamics of vortex formation in trapped Bose-Einstein condensates, Phys. Rev. Lett., 83 (1999), 895-898.
69. R. Carles, "Semi-Classical Analysis for Nonlinear Schrödinger Equations," World Scentific, 2008.
70. R. Carles, P. A. Markowich and C. Sparber, Semiclassical asymptotics for weakly nonlinear Bloch waves, J. Statist. Phys., 117 (2004), 343-375.
71. R. Carles, P. A. Markowich and C. Sparber, On the Gross-Pitaevskii equation for trapped dipolar quantum gases, Nonlinearity, 21 (2008), 2569-2590.
72. L. D. Carr, C. W. Clark and W. P. Reinhardt, Stationary solutions of the one dimensional nonlinear Schrodinger equation I. case of repulsive nonlinearity, Phys. Rev. A, 62 (2000), 063610.
73. T. Cazenave, "Semilinear Schrödinger Equations," Courant Lect. Notes Math., 10, Amer. Math. Soc., Providence, R.I., 2003.
74. M. M. Cerimele, M. L. Chiofalo, F. Pistella, S. Succi and M. P. Tosi, Numerical solution of the Gross-Pitaevskii equation using an explicit finite-difference scheme: An application to trapped Bose-Einstein condensates, Phys. Rev. E, 62 (2000), 1382-1389.
75. M. M. Cerimele, F. Pistella and S. Succi, Particle-inspired scheme for the Gross-Pitaevski equation: An application to Bose-Einstein condensation, Comput. Phys. Comm., 129 (2000), 82-90.
76. Q. Chang, B. Guo and H. Jiang, Finite difference method for generalized Zakharov equations, Math. Comp., 64 (1995), 537-553.
77. S. M. Chang, W. W. Lin and S. F. Shieh, Gauss-Seidel-type methods for energy states of a multi-component Bose-Einstein condensate, J. Comput. Phys., 202 (2005), 367-390.
78. M. L. Chiofalo, S. Succi and M. P. Tosi, Ground state of trapped interacting Bose-Einstein condensates by an explicit imaginary-time algorithm, Phys. Rev. E, 62 (2000), 7438-7444.
79. D. I. Choi and Q. Niu, Bose-Einstein condensation in an optical lattice, Phys. Rev. Lett., 82 (1999), 2022-2025.
80. E. A. Cornell and C. E. Wieman, Nobel Lecture: Bose-Einstein condensation in a dilute gas, the first 70 years and some recent experiments, Rev. Mod. Phys., 74 (2002), 875-893.
81. M. Correggi, P. Florian, N. Rougerie and J. Yngvason, Rotating superfluids in anharmonic traps: From vortex lattices to giant vortices, Phys. Rev. A, 84 (2011), 053614.
82. M. Correggi, N. Rougerie and J. Yngvason, The transition to a giant vortex phase in a fast rotating Bose-Einstein condensate, Comm. Math. Phys., 303 (2011), 451-508.
83. M. Correggi and J. Yngvason, Energy and vorticity in fast rotating Bose-Einstein condensates, J. Phys. A, 41 (2008), pp.44.
84. F. Dalfovo, S. Giorgini, L. P. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71 (1999), 463-512.
85. M. J. Davis and P. B. Blakie, Critical temperature of a trapped Bose gas: Comparison of theory and experiment, Phys. Rev. Lett., 96 (2006), 060404.
86. 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), 3969-3973.
87. M. J. Davis, S. A. Morgan and K. Burnett, Simulations of Bose-fields at finite temperature, Phys. Rev. Lett., 87 (2001), 160402.
88. A. Debussche and E. Faou, Modified energy for split-step methods applied to the linear Schrödinger equations, SIAM J. Numer. Anal., 47 (2009), 3705-3719.
89. C. M. Dion and E. Cances, Spectral method for the time-dependent Gross-Pitaevskii equation with a harmonic trap, Phys. Rev. E, 67 (2003), 046706.
90. R. J. Dodd, Approximate solutions of the nonlinear Schrödinger equation for ground and excited sates for Bose-Einstein condensates, J. Res. Natl. Inst. Stand. Technol., 101 (1996), 545-552.
91. Q. Du, Numerical computations of quantized vortices in Bose-Einstein condensate, in "Recent Progress in Computational and Applied PDEs" (eds. T. Chan et. al.), Kluwer Academic Publisher, (2002), 155-168.
92. Q. Du and F. H. Lin, Numerical approximations of a norm preserving gradient flow and applications to an optimal partition problem, Nonlinearity, 22 (2009), 67-83.
93. M. Edwards and K. Burnett, Numerical solution of the nonlinear Schrödinger equation for small samples of neutral atoms, Phys. Rev. A, 51 (1995), 101103.
94. A. Einstein, Quantentheorie des einatomigen idealen gases, Sitzungsberichte der Preussischen Akademie der Wissenschaften, 22 (1924), 261-267.
95. A. Einstein, Quantentheorie des einatomigen idealen gases, zweite abhandlung, Sitzungsberichte der Preussischen Akademie der Wissenschaften, 1 (1925), 3-14.
96. L. Erdos, B. Schlein and H. T. Yau, Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate, Ann. Math., 172 (2010), 291-370.
97. A. L. Fetter, Rotating trapped Bose-Einstein condensates, Rev. Mod. Phys., 81 (2009), 647-691.
98. A. L. Fetter and A. A. Svidzinsky, Vortices in trapped dilute Bose-Einstein condensate, J. Phys.: Condens. Matter, 13 (2001), 135-194.
99. D. G. Fried, T. C. Killian, L. Willmann, D. Landhuis, S. C. Moss, D. Kleppner and T. J. Greytak, Bose-Einstein condensation of atomic hydrogen, Phys. Rev. Lett, 81 (1998), 3811.
100. J. J. Garcia-Ripoll and V. M. Perez-Garcia, Optimizing Schrödinger functional using Sobolev gradients: Applications to quantum mechanics and nonlinear optics, SIAM J. Sci. Comput., 23 (2001), 1315-1333.
101. J. J. Garcia-Ripoll, V. M. Perez-Garcia and V. Vekslerchik, Construction of exact solutions by spatial translations in inhomogeneous nonlinear Schrödinger equations, Phys. Rev. E, 64 (2001), 056602.
102. S. A. Gardiner, D. Jaksch, R. Dum, J. I. Cirac and P. Zoller, Nonlinear matter wave dynamics with a chaotic potential, Phys. Rev. A, 62 (2000), 023612.
103. I. Gasser and P. A. Markowich, Quantum hydrodynamics, Winger transforms and the classical limit, Assymptot. Anal., 14 (1997), 97-116.
104. P. Gerard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 50 (1997), 321-377.
105. S. Giorgini, L. P. Pitaevskii and S. Stringari, Theory of ultracold atomic Fermi gases, Rev. Mod. Phys., 80 (2008), 1215-1274.
106. R. T. Glassey, Convergence of an energy-preserving scheme for the Zakharov equations in one space dimension, Math. Comp., 58 (1992), 83-102.
107. A. Griesmaier, J. Werner, S. Hensler, J. Stuhler and T. Pfau, Bose-Einstein condensation of Chromium, Phys. Rev. Lett., 94 (2005), 160401.
108. A. Griffin, T. Nikuni and E. Zaremba, "Bose-Condensed Gases at Finite Temperatures," Cambridge University Press, 2009.
109. E. P. Gross, Structure of a quantized vortex in boson systems, Nuovo. Cimento., 20 (1961), 454-457.
110. Paul Lee Halkyard, "Dynamics in Cold Atomic Gases: Resonant Behaviour of the Quantum Delta-Kicked Accelerator and Bose-Einstein Condensates in Ring Traps," Ph.D Thesis, Durham University, 2010.
111. C. Hao, L. Hsiao and H.-L. Li, Global well-posedness for the Gross-Pitaevskii equation with an angular momentum rotational term, Math. Methods Appl. Sci., 31 (2008), 655-664.
112. C. Hao, L. Hsiao and H.-L. Li, Global well-posedness for the Gross-Pitaevskii equation with an angular momentum rotational term in three dimensions, J. Math. Phys., 48 (2007), 102105.
113. D. S. Hall, M. R. Matthews, J. R. Ensher, C. E. Wieman and E. A. Cornell, Dynamics of component separation in a binary mixture of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 1539-1542.
114. R. H. Hardin and F. D. Tappert, Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations, SIAM Rev. Chronicle, 15 (1973), pp. 423.
115. N. Hayashi and T. Ozawa, Remarks on nonlinear Schrödinger equations in one space dimension, Differ. Integral Equ., 2 (1994), 453-461.
116. C. E. Hecht, The possible superfluid behaviour of hydrogen atom gases and liquids, Physica, 25 (1959), 1159-1161.
117. T. L. Ho, Spinor Bose condensates in optical traps, Phys. Rev. Lett., 81 (1998), 742-745.
118. M. Holthaus, Towards coherent control of a Bose-Einstein condensate in a double well, Phys. Rev. A, 64 (2001), 011601.
119. R. Ignat and V. Millot, The critical velocity for vortex existence in a two-dimensional rotating Bose-Einstein condensate, J. Funct. Anal., 233 (2006), 260-306.
120. R. Ignat and V. Millot, Energy expansion and vortex location for a two-dimensional rotating Bose-Einstein condensate, Rev. Math. Phys., 18 (2006), 119-162.
121. D. Jaksch, S. A. Gardiner, K. Schulze, J. I. Cirac and P. Zoller, Uniting Bose-Einstein condensates in optical resonators, Phys. Rev. Lett., 86 (2001), 4733-4736.
122. S. Jin, C. D. Levermore and D. W. McLaughlin, The semiclassical limit of the defocusing nonlinear Schrödinger hierarchy, Comm. Pure Appl. Math., 52 (1999), 613-654.
123. G. Karrali and C. Sourdis, The ground state of a Gross-Pitaevskii energy with general potential in the Thomas-Fermi limit, preprint
124. Y. Kawaguchi and M. Ueda, Spinor Bose-Einstein condensates, Phys. Rep., in press.
125. C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for quasi-linear Schrödinger equations, Invent. Math., 158 (2004), 343-388.
126. W. Ketterle, Nobel lecture: When atoms behave as waves: Bose-Einstein condensation and the atom laser, Rev. Mod. Phys., 74 (2002), 1131-1151.
127. A. Klein, D. Jaksch, Y. Zhang and W. Bao, Dynamics of vortices in weakly interacting Bose-Einstein condensates, Phys. Rev. A, 76 (2007), article 043602.
128. I. Kyza, C. Makridakis and M. Plexousakis, Error control for time-splitting spectral approximations of the semiclassical Schr, arXiv:1205.5997v2.dinger equation, IMA J. Numer. Anal., 31 (2011), 416-441.
129. C. K. Law, H. Pu and N. P. Bigelow, Quantum spins mixing in spinor Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 5257-5261.
130. A. J. Leggett, Bose-Einstein condensation in the alkali gases: Some fundamental concepts, Rev. Mod. Phys., 73 (2001), 307-356.
131. E. H. Lieb and M. Loss, "Analysis," Graduate Studies in Mathematics, Amer. Math. Soc., 2nd ed., 2001.
132. E. H. Lieb and R. Seiringer, Derivation of the Gross-Pitaevskii equation for rotating Bose gases, Comm. Math. Phys., 264 (2006), 505-537.
133. E. H. Lieb, R. Seiringer, J. P. Solovej and J. Yngvason, "The Mathematics of the Bose Gas and its Condensation," Oberwolfach Seminars 34, Birkhäuser Verlag, Basel, 2005.
134. E. H. Lieb, R. Seiringer and J. Yngvason, Bosons in a trap: A rigorous derivation of the Gross-Pitaevskii energy functional, Phys. Rev. A, 61 (2000), 043602.
135. E. H. Lieb and J. P. Solovej, Ground state energy of the two-component charged Bose gas, Comm. Math. Phys., 252 (2004), 485-534.
136. Fong Yin Lim, "Analytical and Numerical Studies of Bose-Einstein Condensates," Ph.D Thesis, National Universtiy of Singapore, 2008.
137. H. L. Liu and C. Sparber, Rigorious derivation of the hydrodynamical equations for rotating superfluids, Math. Models Methods Appl. Sci., 18 (2008), 689-706.
138. W. M. Liu, B. Wu and Q. Niu, Nonlinear effects in interference of Bose-Einstein condensates, Phys. Rev. Lett., 84 (2000), 2294-2297.
139. F. London, The λ-phenomenon of liquid helium and the Bose-Einstein degeneracy, Nature, 141 (1938), 643-644.
140. M. Lu, N. Q. Burdick, S.-H. Youn and B. L. Lev, A strongly dipolar Bose-Einstein condensate of Dysprosium, Phy. Rev. Lett., 107 (2011), 190401.
141. C. Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations, Math. Comp., 77 (2008), 2141-2153.
142. K. W. Madison, F. Chevy, W. Wohlleben and J. Dalibard, Vortex formation in a stirred Bose-Einstein condensate, Phys. Rev. Lett., 84 (2000), 806-809.
143. 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 (1999), 2498-2501.
144. C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell and C. E. Wieman, Production of two overlapping Bose-Einstein condensates by sympathetic cooling, Phys. Rev. Lett., 78 (1997), 586-589.
145. G. J. Milburn, J. Corney, E. M. Wright and D. F. Walls, Quantum dynamics of an atomic Bose-Einstein condensate in a double-well potential, Phys. Rev. A, 55 (1997), 4318.
146. B. Min, T. Li, M. Rosenkranz and W. Bao, Subdiffusive spreading of a Bose-Einstein condensate in random potentials, Phys. Rev. A, 86 (2012), article 053612.
147. O. Morsch and M. Oberthaler, Dynamics of Bose-Einstein condensates in optical lattices, Rev. Mod. Phys., 78 (2006), 179-215.
148. C. Neuhauser and M. Thalhammer, On the convergence of splitting methods for linear evolutionary Schrödinger equations involving an unbounded potential, BIT, 49 (2009), 199-215.
149. R. Ozeri, N. Katz, J. Steinhauer and N. Davidson, Colloquium: Bulk Bogoliubov excitations in a Bose-Einstein condensate, Rev. Mod. Phys., 77 (2005), 187-205.
150. N. G. Parker, C. Ticknor, A. M. Martin and D. H. J. O'Dell1, Structure formation during the collapse of a dipolar atomic Bose-Einstein condensate, Phys. Rev. A, 79 (2009), 013617.
151. C. J. Pethick and H. Smith, "Bose-Einstein Condensation in Dilute Gases," Cambridge University Press, 2002.
152. L. P. Pitaevskii, Vortex lines in an imperfect Bose gas, Soviet Phys. JETP, 13 (1961), 451-454.
153. L. P. Pitaevskii and S. Stringari, "Bose-Einstein Condensation," Clarendon Press, Oxford, 2003.
154. A. Posazhennikova, Colloquium: Weakly interacting, dilute Bose gases in 2D, Rev. Mod. Phys., 78 (2006), 1111-1134.
155. J. L. Roberts, N. R. Claussen, S. L. Cornish and C. E. Wieman, Magnetic field dependence of ultracold inelastic collisions near a Feshbach resonance, Phys. Rev. Lett., 85 (2000), 728-731.
156. M. P. Robinson, G. Fairweather and B. M. Herbst, On the numerical solution of the cubic Schrödinger equation in one space variable, J. Comput. Phys., 104 (1993), 277-284.
157. S. Ronen, D. C. E. Bortolotti and J. L. Bohn, Bogoliubov modes of a dipolar condensate in a cylindrical trap, Phys. Rev. A, 74 (2006), 013623.
158. M. Rosenkranz, D. Jaksch, F. Y. Lim and W. Bao, Self-trapping of Bose-Einstein condensate expanding into shallow optical lattices, Phys. Rev. A, 77 (2008), 063607.
159. N. Rougerie, Vortex rings in fast rotating Bose-Einstein condensates, Arch. Ration. Mech. Anal., 203 (2012), 69-135.
160. P. A. Ruprecht, M. J. Holland, K. Burnett and M. Edwards, Time-dependent solution of the nonlinear Schrödinger equation for Bose-condensed trapped neutral atoms, Phys. Rev. A, 51 (1995), 4704-4711.
161. C. Ryu, M. F. Andersen, P. Cladé, Vasant Natarajan, K. Helmerson and W. D. Phillips, Observation of persistent flow of a Bose-Einstein condensate in a toroidal trap, Phys. Rev. Lett., 99 (2007), 260401.
162. H. Saito and M. Ueda, Intermittent implosion and pattern formation of trapped Bose-Einstein condensates with an attractive interaction, Phys. Rev. Lett., 86 (2001), 1406-1409.
163. J. A. Sanders, F. Verhulst and J. Murdock, "Averaging Methods in Nonlinear Dynamical Systems," 2nd edition, Appl. Math. Sci., 59, Springer, 2007.
164. L. Santos, G. Shlyapnikov, P. Zoller and M. Lewenstein, Bose-Einstein condesation in trapped dipolar gases, Phys. Rev. Lett., 85 (2000), 1791-1797.
165. R. Seiringer, Gross-Pitaevskii theory of the rotating Bose gas, Comm. Math. Phys., 229 (2002), 491-509.
166. J. Shen, Stable and efficient spectral methods in unbounded domains using Laguerre functions, SIAM J. Numer. Anal., 38 (2000), 1113-1133.
167. J. Shen and T. Tang, "Spectral and High-Order Methods with Applications," Science Press, Beijing, 2006.
168. J. Shen, T. Tang and L.-L. Wang, "Spectral Methods. Algorithms, Analysis and Applications," Springer, Heidelberg, 2011.
169. J. Shen and Z.-Q. Wang, Error analysis of the Strang time-splitting Laguerre-Hermite/Hermite collocation methods for the Gross-Pitaevskii equation, J. Found. Comput. Math., to appear.
170. I. F. Silvera and J. T. M. Walraven, Stabilization of atomic Hydrogen at low temperature, Phys. Rev. Lett., 44 (1980), 164-168.
171. T. P. Simula, A. A. Penckwitt and R. J. Ballagh, Giant vortex lattice deformation in rapidly rotating Bose-Einstein condensates, Phys. Rev. Lett., 92 (2004), 060401.
172. C. Sparber, Effective mass theorems for nonlinear Schrödinger equations, SIAM J. Appl. Math., 66 (2006), 820-842.
173. D. M. Stamper-Kurn, A. P. Chikkatur, A. Görlitz, S. Inouye, S. Gupta, D. E. Pritchard and W. Ketterle, Excitation of phonons in a Bose-Einstein condensate by light scattering, Phys. Rev. Lett., 83 (1999), 2876-2879.
174. G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal.,5 (1968), 506-517.
175. R. S. Strichartz, Restriction of Fourier transform to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705-714.
176. C. Sulem and P. L. Sulem, "The Nonlinear Schrödinger Equation, Self-focusing and Wave Collapse," Springer-Verlag, New York, 1999.
177. G. Szegö, "Orthogonal Polynomials," 4th edition, Amer. Math. Soc. Colloq. Publ. 23, AMS, Providence, R.I., 1975.
178. T. R. Taha and M. J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations, II. Numerical, nonlinear Schrödinger equation, J. Comput. Phys., 55 (1984), 203-230.
179. M. Thalhammer, High-order exponential operator splitting methods for time-dependent Schrödinger equations, SIAM J. Numer. Anal., 46 (2008), 2022-2038.
180. V. Thomée, "Galerkin Finite Element Methods for Parabolic Problems," Springer-Verlag, Berlin, Heidelberg, 1997.
181. I. Tikhonenkov, B. A. Malomed and A. Vardi, Anisotropic solitons in dipolar Bose-Einstein condensates, Phys. Rev. Lett., 100 (2008), 090406.
182. S. Utsunomiya, L. Tian, G. Roumpos, C. W. Lai, N. Kumada, T. Fujisawa, M. Kuwata-Gonokami, A. Löffler, S. Höfling, A. Forchel and Y. Yamamoto, Observation of Bogoliubov excitations in exciton-polariton condensates, Nature Phys., 4 (2008), 700-705.
183. Hanquan Wang, "Quantized Vortices States and Dyanmics in Bose-Einstein Condensates," PhD Thesis, National University of Singapore, 2006
184. H. Wang, A time-splitting spectral method for coupled Gross-Pitaevskii equations with applications to rotating Bose-Einstein condensates, J. Comput. Appl. Math., 205 (2007), 88-104.
185. H. Wang, An efficient numerical method for computing dynamics of spin F = 2 Bose-Einstein condensates, J. Comput. Phys., 230 (2011), 6155-6168.
186. H. Wang and W. Xu, An efficient numerical method for simulating the dynamics of coupling Bose-Einstein condensates in optical resonators, Comput. Phys. Comm., 182 (2011), 706- 718.
187. M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576.
188. J. Williams, R. Walser, J. Cooper, E. Cornell and M. Holland, Nonlinear Josephson-type oscillations of a driven two-component Bose-Einstein condensate, Phys. Rev. A, 59 (1999), R31-R34.
189. B. Xiong, J. Gong, H. Pu, W. Bao and B. Li, Symmetry breaking and self-trapping of a dipolar Bose-Einstein condensate in a double-well potential, Phys. Rev. A, 79 (2009), 013626.
190. S. Yi and L. You, Trapped atomic condensates with anisotropic interactions, Phys. Rev. A, 61 (2000), 041604(R).
191. S. Yi and L. You, Expansion of a dipolar condensate, Phys. Rev. A, 67 (2003), 045601.
192. H. Yoshida, Construction of higher order symplectic integrators, Phys. Lett. A, 150 (1990), 262-268.
193. E. Zaremba, T. Nikuni and A. Griffin, Dynamics of trapped Bose gases at finite temperature, J. Low Temp. Phys., 116 (1999), 277.
194. R. Zeng and Y. Zhang, Efficiently computing vortex lattices in fast rotating Bose-Einstein condensates, Comput. Phys. Commun., 180 (2009), 854-860.
195. P. Zhang, "Wigner Measure and Semiclassical Limits of Nonlinear Schrödinger Equations," Courant Lect. Notes Math., 17, Amer. Math. Soc., Providence, R.I., 2008.
196. Yanzhi Zhang, "Mathematical Analysis and Numerical Simulation for Bose-Einstein Condensation," PhD Thesis, National University of Singapore, 2006
197. Y. Zhang and W. Bao, Dynamics of the center of mass in rotating Bose-Einstein condensates, Appl. Numer. Math., 57 (2007), 697-709.
198. Y. Zhang, W. Bao and H. Li, Dynamics of rotating two-component Bose-Einstein condensates and its efficient computation, Phys. D, 234 (2007), 49-69.