Mathematical theory and numerical methods for gross pitaevskii equations and applications

MATHEMATICAL THEORY AND NUMERICAL METHODS FOR GROSS-PITAEVSKII EQUATIONS AND APPLICATIONS CAI YONGYONG (M.Sc., Peking University) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 2011 Acknowledgements It is a great pleasure for me to take this opportunity to thank those who made this thesis possible. First and foremost, I would like to express my heartfelt gratitude to my supervisor Prof. Weizhu Bao, for his encouragement, patient guidance, generous support and invaluable advice. He has taught me a lot in both research and life. I would also like to thank my other collaborators for their contribution to the work: Prof. Naoufel Ben Abdallah , Dr. Hanquan Wang, Dr. Zhen Lei and Dr. Matthias Rosenkranz. Many thanks to Naoufel Ben Abdallah for his kind hospitality during my visit in Toulouse. Special thanks to Yanzhi for reading the draft. I also want to thank my family for their unconditional support. The last but no least, I would like to thank all the colleagues, friends and staffs here in Department of Mathematics, National University of Singapore. ii Contents Introduction 1.1 The Gross-Pitaevskii equation . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Ground state and dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Existing results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Purpose of study and structure of thesis . . . . . . . . . . . . . . . . . . . . Gross-Pitaevskii equation for degenerate dipolar quantum gas 11 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Analytical results for ground states and dynamics . . . . . . . . . . . . . . . 14 2.2.1 Existence and uniqueness for ground states . . . . . . . . . . . . . . 15 2.2.2 Analytical results for dynamics . . . . . . . . . . . . . . . . . . . . . 20 2.3 A numerical method for computing ground states . . . . . . . . . . . . . . . 22 2.4 A time-splitting pseudospectral method for dynamics . . . . . . . . . . . . . 28 2.5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.5.1 Comparison for evaluating the dipolar energy . . . . . . . . . . . . . 29 2.5.2 Ground states of dipolar BECs . . . . . . . . . . . . . . . . . . . . . 31 2.5.3 Dynamics of dipolar BECs . . . . . . . . . . . . . . . . . . . . . . . 32 iii Contents iv Dipolar Gross-Pitaevskii equation with anisotropic confinement 36 3.1 Lower dimensional models for dipolar GPE . . . . . . . . . . . . . . . . . . 36 3.2 Results for the quasi-2D equation I . . . . . . . . . . . . . . . . . . . . . . . 38 3.3 3.4 3.5 3.6 3.7 3.2.1 Existence and uniqueness of ground state . . . . . . . . . . . . . . . 39 3.2.2 Well-posedness for dynamics . . . . . . . . . . . . . . . . . . . . . . 44 Results for the quasi-2D equation II . . . . . . . . . . . . . . . . . . . . . . 47 3.3.1 Existence and uniqueness of ground state . . . . . . . . . . . . . . . 47 3.3.2 Existence results for dynamics . . . . . . . . . . . . . . . . . . . . . 52 Results for the quasi-1D equation . . . . . . . . . . . . . . . . . . . . . . . . 57 3.4.1 Existence and uniqueness of ground state . . . . . . . . . . . . . . . 58 3.4.2 Well-posedness for dynamics . . . . . . . . . . . . . . . . . . . . . . 59 Convergence rate of dimension reduction . . . . . . . . . . . . . . . . . . . . 60 3.5.1 Reduction to 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.5.2 Reduction to 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.6.1 Numerical method for the quasi-2D equation I . . . . . . . . . . . . 67 3.6.2 Numerical method for the quasi-1D equation . . . . . . . . . . . . . 69 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Dipolar Gross-Pitaevskii equation with rotational frame 75 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.2 Analytical results for ground states . . . . . . . . . . . . . . . . . . . . . . . 78 4.3 A numerical method for computing ground states of (4.11) . . . . . . . . . . 80 4.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Ground states of coupled Gross-Pitaevskii equations 85 5.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.2 Existence and uniqueness results for the ground states . . . . . . . . . . . . 89 5.2.1 For the case with optical resonator, i.e. problem (5.12) . . . . . . . . 89 5.2.2 For the case without optical resonator and Josephson junction, i.e. problem (5.14) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.3 Properties of the ground states . . . . . . . . . . . . . . . . . . . . . . . . . 101 Contents 5.4 5.5 v Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.4.1 Continuous normalized gradient flow and its discretization . . . . . . 105 5.4.2 Gradient flow with discrete normalization and its discretization . . . 108 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation 122 6.1 The equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 6.2 Finite difference methods and main results . . . . . . . . . . . . . . . . . . . 124 6.2.1 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.2.2 Main error estimate results . . . . . . . . . . . . . . . . . . . . . . . 126 6.3 Error estimates for the SIFD method . . . . . . . . . . . . . . . . . . . . . . 128 6.4 Error estimates for the CNFD method . . . . . . . . . . . . . . . . . . . . . 138 6.5 Extension to other cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6.6 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Uniform error estimates of finite difference methods for the nonlinear Schr¨ odinger equation with wave operator 153 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 7.2 Finite difference schemes and main results . . . . . . . . . . . . . . . . . . . 156 7.2.1 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 7.2.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 7.3 Convergence of the SIFD scheme . . . . . . . . . . . . . . . . . . . . . . . . 161 7.4 Convergence of the CNFD scheme . . . . . . . . . . . . . . . . . . . . . . . 173 7.5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Concluding remarks and future work 185 A Proof of the equality (2.15) 188 B Derivation of quasi-2D equation I (3.4) 190 C Derivation of quasi-1D equation (3.10) 192 Contents vi D Outline of the convergence between NLSW and NLSE 194 Bibliography 196 List of Publications 212 Summary Gross-Pitaevskii equation (GPE), first derived in early 1960s, is a widely used model in different subjects, such as quantum mechanics, condensed matter physics, nonlinear optics etc. Since 1995, GPE has regained considerable research interests due to the experimental success of Bose-Einstein condensates (BEC), which can be well described by GPE at ultra-cold temperature. The purpose of this thesis is to carry out mathematical and numerical studies for GPE. We focus on the ground states and the dynamics of GPE. The ground state is defined as the minimizer of the energy functional associated with the corresponding GPE, under the constraint of total mass (L2 norm) being normalized to 1. For the dynamics, the task is to solve the Cauchy problem for GPE. This thesis mainly contains three parts. The first part is to investigate the dipolar GPE modeling degenerate dipolar quantum gas. For ground states, we prove the existence and uniqueness as well as non-existence. For dynamics, we discuss the well-posedness, possible finite time blow-up and dimension reduction. Convergence for this dimension reduction has been established in certain regimes. Efficient and accurate numerical methods are proposed to compute the ground states and the dynamics. Numerical results show the efficiency and accuracy of the numerical methods. The second part is devoted to the coupled GPEs modeling a two component BEC. We show the existence and uniqueness as well as non-existence and limiting behavior of the ground states in different parameter regimes. Efficient and accurate numerical methods vii Summary are designed to compute the ground states. Examples are shown to confirm the analytical analysis. The third part is to understand the convergence of the finite difference discretizations for GPE. We prove the optimal convergence rates for the conservative Crank-Nicolson finite difference discretizations (CNFD) and the semi-implicit finite difference discretizations (SIFD) for rotational GPE, in two and three dimensions. We also consider the nonlinear Schr¨ odinger equation perturbed by the wave operator, where the small perturbation causes high oscillation of the solution in time. This high oscillation brings significant difficulties in proving uniform convergence rates for CNFD and SIFD, independent of the perturbation. We overcome the difficulties and obtain uniform error bounds for both CNFD and SIFD, in one, two and three dimensions. Numerical results confirm our theoretical analysis. viii Notations t time i imaginary unit x spatial variable Rd d dimensional Euclidean space ψ := ψ(x, t) complex wave-function Planck constant ∇ gradient ∇2 = ∇ · ∇, ∆ Laplace operator c¯ conjugate of c Re(c) real part of c Im(c) imaginary part of c Lz = −i(x∂y − y∂x ) z-component of angular momentum u p := u Lp (Rd ) Lp (p ∈ [1, ∞]) norm of function u(x), where there is no confusion about d fˆ(ξ) := Rd f (x)e−ix·ξ dx Fourier transform of f (x) ix Chapter Introduction The Gross-Pitaevskii equation (GPE), also known as the cubic nonlinear Schr¨ odinger equation (NLSE), has various physics applications, such as quantum mechanics, condensate matter physics, nonlinear optics, water waves, etc. The equation was first developed to describe identical bosons by Eugene P. Gross [72] and Lev Petrovich Pitaevskii [116] in 1961, independently. Later, GPE has been found various applications in other areas, known as the cubic NLSE. Since 1995, the Gross-Pitaevskii theory of boson particles has regained great interest due to the successful experimental treatment of the dilute boson gas, which resulted in the remarkable discovery of Bose-Einstein condensate (BEC) [7, 36, 52]. Now, BEC has become one of the hottest research topics in physics, and motivates numerous mathematical and numerical studies on GPE. 1.1 The Gross-Pitaevskii equation Many different physical applications lead to the Gross-Pitaevskii equation (GPE). For example, in BEC experiments, near absolute zero temperature, a large portion of the dilute atomic gas confined in an external trapping potential occupies the same lowest energy state and forms condensate. At temperature T much lower than the critical temperature Tc , using mean field approximation for this dilute many-body system, BEC can be described by a macroscopic wave function ψ(x, t), governed by GPE in the dimensionless form [16, 18, 117] i∂t ψ(x, t) = − ∇2 ψ(x, t) + Vd (x)ψ(x, t) + βd |ψ(x, t)|2 ψ(x, t), x ∈ Rd , d = 1, 2, 3, (1.1) Bibliography [19] W. Bao, S. Jin, and P. A. Markowich, On time-splitting spectral approximation for the Schr¨ odinger equation in the semiclassical regime, J. Comput. Phys., 175 (2002), pp. 487–524. [20] W. Bao, H. 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), pp. 3685–3711. [21] 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), pp. 1925–1948. [22] W. Bao, P. A. Markowich, C. Schmeiser and R. M. Weish¨ aupl, On the Gross-Pitaevskii equation with strongly anisotropic confinement: formal asymptotics and numerical experiments, Math. Models Meth. Appl. Sci., 15 (2005), pp. 767–782. [23] 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), pp. 9778–9793. [24] W. Bao and W. Tang, Ground state solution of Bose-Einstein condensate by directly minimizing the energy functional, J. Comput. Phys., 187 (2003), pp. 230–254. [25] 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), pp. 612–626. [26] 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), pp. 2177–2200. [27] W. Bao, H. Wang and P. A. Markowich, Ground state, symmetric and central vortex state in rotating Bose-Einstein condensate, Comm. Math. Sci., (2005), pp. 57–88. [28] W. Bao and Y. Zhang, Dynamics of the ground state and central vortex states in BoseEinstein condensation, Math. Models Meth. Appl. Sci., 15 (2005), pp. 1863–1896. 198 Bibliography [29] N. Ben Abdallah, F. M´ehats, C. Schmeiser, and R. M. Weish¨ aupl, The nonlinear Schr¨ odinger equation with strong anisotropic harmonic potential, SIAM J. Math. Anal., 37 (2005), pp. 189–199. [30] N. Ben Abdallah, F. Castella, F. M´ehats, Time averaging for the strongly confined nonlinear Schr¨ odinger equation, using almost periodicity, J. Diff. Equ., 245 (2008), pp. 154–200. [31] L. Berg´e and T. Colin, A singular perturbation problem for an enveloppe equation in plasma physics, Physica D, 84 (1995), pp. 437–459. [32] C. Besse, B. Bid´egaray and S. Descombes, Order estimates in time of splitting methods for the nonlinear Schr¨ odinger equation, SIAM J. Numer. Anal., 40 (2002), pp. 26–40. [33] P. B. Blakie, C. Ticknor, A. S. Bradley, A. M. Martin, M. J. Davis and Y. Kawaguchi, Numerical method for evolving the dipolar projected Gross-Pitaevskii equation, Phys. Rev. E, 80 (2009), aritcle 016703. [34] S. N. Bose, Plancks gesetz und lichtquantenhypothese, Zeitschrift fr Physik, (1924), pp. 178. [35] J. Bourgain, Global solutions of nonlinear Schr¨ odinger equations, Colloquium publications, Amer. Math. Soc., 1999. [36] 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), pp. 1687–1690. [37] A. G. Bratsos, On the numerical solution of the nonlinear cubic Schr¨ odinger equation, Int. J. Pure Appl. Math. Sci., (2005), pp. 217–226. [38] L. Cafferelli and F. H. Lin, An optimal partition problem for eigenvalues, J. Sci. Comput., 31 (2007), pp. 5–18. [39] M. Caliari, A. Ostermann, S. Rainer and M. Thalhammer, A minimisation approach for computing the ground state of Gross-Pitaevskii systems, J. Comput. Phys., 228 (2009), pp. 349–360. 199 Bibliography [40] M. Caliari and M. Squassina, Location and phase segregation of ground and excited states for 2D Gross-Pitaevskii systems, Dynamics of PDE., (2008), pp. 117–137. [41] 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), pp. 895– 898. [42] R. Carles, P. A Markowich and C. Sparber, On the Gross-Pitaevskii equation for trapped dipolar quantum gases, Nonlinearity, 21 (2008), pp. 2569–2590. [43] T. Cazenave, Semilinear Schr¨ odinger equations, (Courant Lecture Notes in Mathematics vol. 10), New York University, Courant Institute of Mathematical Sciences, AMS, 2003. [44] 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), pp. 1382– 1389. [45] M. Cerimele, F. Pistella and S. Succi, Particle-like scheme for the Gross- Pitaevski equation: an application to Bose-Einstein condensation, Comput. Phys. Commun., 129 (2000), pp. 82–90. [46] Q. Chang, B. Guo and H. Jiang, Finite difference method for generalized Zakharov equations, Math. Comp., 64 (1995), pp. 537–553. [47] Q. Chang, E. Jia and W. Sun, Difference schemes for solving the generalized nonlinear Schr¨ odinger equation, J. Comput. Phys., 148 (1999), pp. 397–415. [48] 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), pp. 367–390. [49] S. M. Chang, C. S. Lin, T. C. Lin and W. W. Lin, Segregated nodal domains of two-dimensional multispecies Bose-Einstein condensates, Physica D, 196 (2004), pp. 341–361. 200 Bibliography [50] M. L. Chiofalo, S. Succi and M. P. Tosi, Ground state of trapped interacting BoseEinstein condensates by an explicit imaginary-time algorithm, Phys. Rev. E, 62 (2000), pp. 7438–7444. [51] T. Colin and P. Fabrie, Semidiscretization in time for Schr¨ odinger-waves equations, Discrete Contin. Dyn. Syst., (1998), pp. 671–690. [52] 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. [53] M. J. Davis, S. A. Morgan and K. Burnett, Simulations of Bose-fields at finite temperature, Phys. Rev. Lett., 87 (2001), article 160402. [54] A. Debussche and E. Faou, Modified energy for split-step methods applied to the linear Schr¨ odinger equations, SIAM J. Numer. Anal., 47 (2009), pp. 3705–3719. [55] C. M. Dion and E. Cances, Spectral method for the time-dependent Gross-Pitaevskii equation with a harmonic trap, Phys. Rev. E, 67 (2003), article 046706. [56] R. J. Dodd, Approximate solutions of the nonlinear Schr¨ odinger equation for ground and excited sates for Bose-Einstein condensates, J. Res. Natl. Inst. Stand. Technol., 101 (1996), pp. 545–552. [57] Q. Du and F. H. Lin, Numerical approximations of a norm preserving gradient flow and applications to an optimal partition problem, Nonlinearity, 22 (2009), pp. 67–83. [58] C. Eberlein, S. Giovanazzi, and D. H. J. O′ Dell, Exact solution of the Thomas-Fermi equation for a trapped Bose-Einstein condensate with dipole-dipole interactions, Phys. Rev. A, 71 (2005), article 033618. [59] M. Edwards and K. Burnett, Numerical solution of the nonlinear Schr¨ odinger equation for small samples of neutral atoms, Phys. Rev. A, 51 (1995), article 101103. [60] M. Edwards, P. A. Ruprecht, K. Burnett, R. J. Dodd, and C. W. Clark, Collective excitations of atomic Bose-Einstein condensates, Phys. Rev. Lett., 77 (1996), pp. 1671– 1674, article 009817. 201 Bibliography [61] A. Einstein, Quantentheorie des einatomigen idealen gases, Sitzungsberichte der Preussischen Akademie der Wissenschaften, 22 (1924), pp. 261. [62] A. Einstein, Quantentheorie des einatomigen idealen gases, zweite abhandlung, Sitzungs- berichte der Preussischen Akademie der Wissenschaften, (1925), pp. 3. [63] M. S. Ellio, J. J. Valentini, and D. W. Chandler, Subkelvin cooling NO molecules via ”billiard-like” collisions with argon, Science, 302 (2003), pp. 1940–1943. [64] L. Erd˝ os, B. Schlein, and H. T. Yau, Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate, Ann. Math., 172 (2010), pp. 291–370. [65] D. Funaro, Polynomial approximations of differential equations, Springer-Verlag, Berlin, 1992. [66] S. Giovanazzi, P. Pedri, L. Santos, A. Griesmaier, M. Fattori, T. Koch, J. Stuhler and T. Pfau, Expansion dynamics of a dipolar Bose-Einstein condensate, Phys. Rev. A, 74 (2006), article 013621. [67] R. T. Glassey, Convergence of an energy-preserving scheme for the Zakharov equations in one space dimension, Math. Comp., 58 (1992), pp. 83–102. [68] K. Glaum and A. Pelster, Bose-Einstein condensation temperature of dipolar gas in anisotropic harmonic trap, Phys. Rev. A, 76 (2007), article 023604. [69] K. Go’ral, K. Rzayewski and T. Pfau, Bose-Einstein condensation with magnetic dipole-dipole forces, Phys. Rev. A, 61 (2000), article 051601. [70] K. Go’ral and L. Santos, Ground state and elementary excitations of single and binary Bose-Einstein condensates of trapped dipolar gases, Phys. Rev. A, 66 (2002), article 023613. [71] A. Griesmaier, J. Werner, S. Hensler, J. Stuhler and T. Pfau, Bose-Einstein condensation of Chromium, Phys. Rev. Lett., 94 (2005), article 160401. [72] E. P. Gross, Structure of a quantized vortex in boson systems, Nuovo. Cimento., 20 (1961), pp. 454–457. 202 Bibliography [73] B. Guo and H. Liang, On the problem of numerical calculation for a class of systems of nonlinear Schr¨ odinger equations with wave operator, J. Numer. Methods Comput. Appl., (1983), pp. 176–182. [74] 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), pp. 1539–1542. [75] D. S. Hall, M. R. Matthews, C. E. Wieman and E. A. Cornell, Measurements of relative phase in two-component Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), pp. 1543–1546. [76] C. Hao, L. Hsiao, and H. Li, Global well posedness for the Gross-Pitaevskii equation with an angular momentum rotational term in three dimensions, J. Math. Phys., 48 (2007), article 102105. [77] C. Hao, L. Hsiao and H. Li, Global well posedness for the Gross-Pitaevskii equation with an angular momentum rotational term, Math. Meth. Appl. Sci., 31 (2008), pp. 655–664. [78] 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. [79] N. Hayashi and T. Ozawa, Remarks on nonlinear Schr¨ oinger equations in one space dimension, Differential Integral Equations, (1994), pp. 453–461. [80] T. L. Ho and V. B. Shenoy, Binary mixtures of Bose condensates of alkali atoms, Phys. Rev. Lett., 77 (1996), pp. 3276–3279. [81] J. Hong, Y. Liu, H. Munthe-Kaas and A. Zanna, Globally conservative properties and error estimation of a multi-symplectic scheme for Schr¨ odinger equations with variable coefficients, Appl. Numer. Math., 56 (2006), pp. 814–843. [82] L. I. Ignat and E. Zuazua, Numerical dispersive schemes for the nonlinear Schr¨ odinger equation, SIAM J. Numer. Anal., 47 (2009), pp. 1366–1390. 203 Bibliography [83] 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), pp. 4733–4736. [84] M. Javidi and A. Golbabai, Numerical studies on nonlinear Schr¨ odinger equations by spectral collocation method with preconditioning, J. Math. Anal. Appl., 333 (2007), pp. 1119–1127. [85] T. F. Jiang and W. C. Su, Ground state of the dipolar Bose-Einstein condensate, Phys. Rev. A, 74 (2006), article 063602. [86] O. Karakashian, G. Akrivis, and V. Dougalis, On optimal order error estimates for the nonlinear Schr¨ odinger equation, SIAM J. Numer. Anal., 30 (1993), pp. 377–400. [87] K. Kasamatsu and M. Tsubota, Nonlinear dynamics for vortex formation in a rotating Bose-Einstein condensate, Phys. Rev. A, 67 (2003), article 033610. [88] K. Kasamatsu, M. Tsubota and M. Ueda, Vortex phase diagram in rotating twocomponent Bose-Einstein condensates, Phys. Rev. Lett., 91 (2003), article 150406. [89] P.L. Kelley, Self-focusing of optical beams, Phys. Rev. Lett., 15 (1965), pp. 1005–1008. [90] C. E. Kenig, G. Ponce and L. Vega, Small solutions to nonlinear Schr¨ odinger equations, Ann. Inst. Henry Poincar´e, 10 (1993), pp. 255–288. [91] C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for quasi-linear Schr¨ odinger equations, Invent. Math., 158 (2004), pp. 343–388. [92] M. Klawunn, R. Nath, P. Pedri and L. Santos, Transverse instability of straight vortex lines in dipolar Bose-Einstein condensates, Phys. Rev. Lett., 100 (2008), article 240403. [93] T. Lahaye, J. Metz, B. Fr¨ ohlich, T. Koch, M. Meister, A. Griesmaier, T. Pfau, H. Saito, Y. Kawaguchi and M. Ueda, D-wave collapse and explosion of a dipolar BoseEinstein condensate, Phys. Rev. Lett., 101 (2008), article 080401. [94] R. Landes, On Galerkin’s method in the existence theory of quasilinear elliptic equations, J. Funct. Anal., 39 (1980), pp. 123–148. [95] M. Lees, Approximate solution of parabolic equations, J. Soc. Indust. Appl. Math., (1959), pp. 167–183. 204 Bibliography [96] E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, Amer. Math. Soc., 2nd ed., 2001. [97] E. H. Lieb and R. Seiringer, Derivation of the Gross-Pitaevskii equation for rotating Bose gases, Commun. Math. Phys., 264 (2006), pp. 505–537. [98] 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), article 043602. [99] E. H. Lieb and J. P. Solovej, Ground state energy of the two-component charged Bose gas, Commun. Math. Phys., 252 (2004), pp. 485–534. [100] T. -C. Lin and J. Wei, Ground state of N coupled nonlinear Schr¨ odinger equations in Rn , n ≤ 3, Commun. Math. Phys., 255 (2005), pp. 629–653. [101] T. -C. Lin and J. Wei, Spikes in two-component systems of nonlinear Schr¨ odinger equations with trapping potentials, J. Diff. Equ., 229 (2006), pp. 538–569. [102] Z. Liu, Two-component Bose-Einstein condensates, J. Math. Anal. Appl., 348 (2008), pp. 274–285. [103] C. Lubich, On splitting methods for Schr¨ odinger-Poisson and cubic nonlinear Schr¨ odinger equations, Math. Comp., 77 (2008), pp. 2141–2153. [104] S. Machihara, K. Nakanishi and T. Ozawa, Nonrelativistic limit in the energy space for nonlinear Klein-Gordon equations, Math. Ann., 322 (2002), pp. 603–621. [105] P. A. Markowich, P. Pietra and C. Pohl, Numerical approximation of quadratic observables of Schr¨ odinger-type equations in the semi-classical limit, Numer. Math., 81 (1999), pp. 595–630. [106] M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wiemann, and E. A. Cornell, Vortices in a Bose-Einstein condensate, Phys. Rev. Lett., 83 (1999), pp. 2498–2501. [107] P. Muruganandam and S. K. Adhikari, Bose-Einstein condensation dynamics in three dimensions by the pseudospectral and finite-difference methods, J. Phys. B, 36 (2003), pp. 2501–2513. 205 Bibliography [108] 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), pp. 586–589. [109] R. Nath, P. Pedri and L. Santos, Soliton-soliton scattering in dipolar Bose-Einstein condensates, Phys. Rev. A, 76 (2007), article 013606. [110] C. Neuhauser and M. Thalhammer, On the convergence of splitting methods for linear evolutionary Schr¨ odinger equations involving an unbounded potential, BIT, 49 (2009), pp. 199–215. [111] T. Nikuni, E. Zaremba and A. Griffin, Two-fluid dynamics for a Bose-Einstein condensate out of local equilibrium with the noncondensate, Phys. Rev. Lett., 83 (1999), pp. 10–13. [112] D. H. J. O′ Dell, S. Giovanazzi and C. Eberlein, Exact hydrodynamics of a trapped dipolar Bose-Einstein condensate, Phys. Rev. Lett., 92 (2004), article 250401. [113] D. H. O′ Dell and C. Eberlein, Vortex in a trapped Bose-Einstein condensate with dipole-dipole interactions, Phys. Rev. A, 75 (2007), article 013604. [114] K. Ohannes and M. Charalambos, A space-time finite element method for the nonlinear Schr¨ odinger equation: the continuous Galerkin method, SIAM J. Numer. Anal., 36 (1999), pp. 1779–1807. [115] 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), article 013617. [116] L. P. Pitaevskii, Vortex lines in an imperfect Bose gas, Soviet Phys. JETP, 13 (1961), pp. 451–454. [117] L. P. Pitaevskii and S. Stringari, Bose-Einstein condensation, Oxford University, New York, 2003. [118] P. Pedri and L. Santos, Two-dimensional bright solitons in dipolar Bose-Einstein condensates, Phys. Rev. Lett., 95 (2005), article 200404. 206 Bibliography [119] A. Recati, I. Carusotto, C. Lobo and S. Stringari, Dipole polarizability of a trapped superfluid Fermi gas, Phys. Rev. Lett., 97 (2006), article 190403. [120] B. Reichel and S. Leble, On convergence and stability of a numerical scheme of coupled nonlinear Schr¨ odinger equations, Comput. Math. Appl., 55 (2008), pp. 745– 759. [121] M. P. Robinson, G. Fairweather and B. M. Herbst, On the numerical solution of the cubic Schr¨ odinger equation in one space variable, J. Comput. Phys., 104 (1993), pp. 277–284. [122] 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), article 013623. [123] J. M. Sage, S. Sainis, T. Bergeman and D. DeMille, Optical production of ultracold polar molecules, Phys. Rev. Lett., 94 (2005), article 203001. [124] L. Salasnich, Generalized nonpolynomial Schr¨ odinger equations for matter waves under anisotropic transverse confinement, J. Phys. A: Math. Theor., 42 (2009), article 335205. [125] L. Santos, G. Shlyapnikov, P. Zoller and M. Lewenstein, Bose-Einstein condesation in trapped dipolar gases, Phys. Rev. Lett., 85 (2000), pp. 1791–1797. [126] B. I. Schneider and D. L. Feder, Numerical approach to the ground and excited states of a Bose-Einstein condensed gas confined in a completely anisotropic trap, Phys. Rev. A, 59 (1999), pp. 2232–2242. [127] J. Schneider and A. Schenzle, Output from an atom laser: theory vs. experiment, Appl. Phys. B, 69 (1999), pp. 353–356. [128] J. Schneider and A. Schenzle, Investigations of a two-mode atom-laser model, Phys. Rev. A, 61 (2000), article 053611. [129] A. Y. Schoene, On the nonrelativistic limits of the Klein-Gordon and Dirac equations, J. Math. Anal. Appl., 71 (1979), pp. 36–47. 207 Bibliography [130] R. Seiringer, Gross-Pitaevskii theory of the rotating Bose gas, Commun. Math. Phys. 229 (2002), pp. 491–509. [131] J. Shen and T. Tang, Spectral and high-order methods with applications, Science Press, Beijing, 2006. [132] L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. Math., 118 (1983), pp. 525–571. [133] D. M. Stamper-Kurn, M. R. Andrews, A. P. Chikkatur, S. Inouye, H.-J. Miesner, J. Stenger and W. Ketterle, Optical confinement of a Bose-Einstein condensate, Phys. Rev. Lett., 80 (1998), pp. 2027–2030. [134] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, New Jersey, 1970. [135] G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal., (1968), pp. 505–517. [136] R. S. Strichartz, Restriction of Fourier transform to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), pp. 705–714. [137] S. Succi, Numerical solution of the Schr¨ odinger equation using discrete kinetic theory, Phys. Rev. E, 53 (1996), pp. 1969–1975. [138] S. Succi, Lattice quantum mechanics: an application to Bose-Einstein condensation, Int. J. Mod. Phys., (1998), pp. 1577–1585, article 101142. [139] C. Sulem and P.-L. Sulem, The nonlinear Schr¨ odinger equation, self-focusing and wave collapse, Springer-Verlag, New York, 1999. [140] G. Szeg¨ o, Orthogonal polynomials 4th ed., Amer. Math. Soc. Colloq. Publ. 23, AMS, Providence, RI, 1975. [141] V. I. Talanov, Self-focusing of wave beams in nonlinear media, Sov. Phys. JEPT Lett., (1965), pp. 138–141. 208 Bibliography [142] T. R. Taha and M. J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations, II. numerical, nonlinear Schr¨ odinger equation, J. Comput. Phys., 55 (1984), pp. 203–230. [143] M. Thalhammer, High-order exponential operator splitting methods for timedependent Schr¨ odinger equations, SIAM J. Numer. Anal., 46 (2008), pp. 2022–2038. [144] M. Thalhammer, M. Caliari and C. Neuhauser, High-order time-splitting Hermite and Fourier spectral methods, J. Comput. Phys., 228 (2009), pp. 822–832. [145] V. Thom´ee, Galerkin finite element methods for parabolic problems, Springer-Verlag, Berlin, Heidelberg, 1997. [146] Y. Tian and M. Qin, Explicit symplectic schemes for investigating the evolution of vortices in a rotating Bose-Einstein condensate, Comput. Phys. Commun., 155 (2003), pp. 132–143. [147] C. Ticknor, N. G. Parker, A. Melatos, S. L. Cornish, D. H. J. O’Dell and A. M. Martin, Collapse times of dipolar Bose-Einstein condensates, Phys. Rev. A, 78 (2008), article 061607. [148] I. Tikhonenkov, B. A. Malomed and A. Vardi, Anisotropic solitons in dipolar BoseEinstein condensates, Phys. Rev. Lett., 100 (2008), article 090406. [149] V. A. Trofimov and N. V. Peskov, Comparison of finite-difference schemes for the Gross-Pitaevskii equation, Math. Model. Anal., 14 (2009), pp. 109–126. [150] M. Tsutumi, Nonrelativistic approximation of nonlinear Klein-Gordon equations in two space dimensions, Nonlinear Analysis, (1984), pp. 637–643. [151] H. Wang, Numerical simulations on stationary states for rotating two-component Bose-Einstein condensates, J. Sci. Comput., 38 (2009), pp. 149–163. [152] D. Wang, J. Qi, M. F. Stone, O. Nikolayeva, H. Wang, B. Hattaway, S. D. Gensemer, P. L. Gould, E. E. Eyler and W. C. Stwalley, Photoassociative production and trapping of ultracold KRb molecules, Phys. Rev. Lett., 93 (2004), article 243005. 209 Bibliography [153] H. Wang and W. Xu, An efficient numerical method for simulating the dynamics of coupling Bose-Einstein condensates in optical resonators, Comput. Phys. Commun., 182 (2011), pp. 706–718. [154] T. Wang and L. Zhang, Analysis of some new conservative schemes for nonlinear Schr¨ odinger equation with wave operator, Appl. Math. Comput., 182 (2006), pp. 1780– 1794. [155] M. I. Weinstein, Nonlinear Schr¨ odinger equations and sharp interpolation estimates, Commun. Math. Phys., 87 (1983), pp. 567–576. [156] J. Williams, R. Walser, J. Cooper, E. Cornell, and M. Holland, Nonlinear Josephsontype oscillations of a driven two-component Bose-Einstein condensate, Phys. Rev. A, 59 (1999), article R31–R34. [157] R. M. Wilson, S. Ronen and J. L. Bohn, Stability and excitations of a dipolar BoseEinstein condensate with a vortex, Phys. Rev. A, 79 (2009), article 013621. [158] R. M. Wilson, S. Ronen, J. L. Bohn, and H. Pu, Manifestations of the roton mode in dipolar Bose-Einstein condensates, Phys. Rev. Lett., 100 (2008), article 245302. [159] J. Xin, Modeling light bullets with the two-dimensional sine-Gordon equation, Physica D, 135 (2000), pp. 345–368. [160] 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), article 013626. [161] S. Yi and H. Pu, Vortex structures in dipolar condensates, Phys. Rev. A, 73 (2006), article 061602. [162] S. Yi and L. You, Trapped atomic condensates with anisotropic interactions, Phys. Rev. A, 61 (2000), article 041604. [163] S. Yi and L. You, Trapped condensates of atoms with dipole interactions, Phys. Rev. A, 63 (2001), article 053607. 210 Bibliography [164] S. Yi and L. You, Expansion of a dipolar condensate, Phys. Rev. A, 67 (2003), article 045601. [165] S. Yi and L. You, Calibrating dipolar interaction in an atomic condensate, Phys. Rev. Lett., 92 (2004), article 193201. [166] Z. Rong and Y. Zhang, Efficiently computing vortex lattices in rapid rotating BoseEinstein condensates, Comput. Phys. Commun., 180 (2009) pp. 854–860. [167] Y. Zhang, W. Bao, and H. Li, Dynamics of rotating two-component Bose-Einstein condensates and its efficient computation, Physica D, 234 (2007), pp. 49–69. [168] J. Zhang and H. Zhai, Vortex lattice in planar Bose-Einstein condendsates with dipolar interactions, Phys. Rev. Lett., 95 (2005), article 200403. 211 List of Publications [1] “Second order averaging for the nonlinear Schrodinger equation with strong anisotropic potential” (with Naoufel Ben Abdallah, Francois Castella and Florian M´ehats), Kinet. Relat. Models., Vol. 4, no. 4, pp. 831-856, 2011. [2] “Mean-field regime of trapped dipolar Bose-Einstein condensates in one and two dimensions” (with Weizhu Bao, Matthias Rosenkranz and Zhen Lei), Phys. Rev. A, Vol. 82, 2010, article 043623. [3] “Ground states of two-component Bose-Einstein condensates with an internal atomic Josephson junction” (with Weizhu Bao), East Asia Journal on Applied Mathematics, Vol. 1, pp. 49–81, 2010. [4] “Efficient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates” (with Weizhu Bao and Hanquan Wang), J. Comput. Phys., Vol. 229, pp. 7874–7892, 2010. [5] “Existence and uniqueness of weak solutions for a non-uniformly parabolic equation” (with Shulin Zhou), J. Funct. Anal., 257, no. 10, pp. 3021–3042, 2009. [6] “Lq interior estimates of the gradient for weak solutions of quasilinear elliptic equations” (with Fengping Yao), Acta Math. Sin. (Engl. Ser.), 23, no. 12, pp. 2225– 2234, 2007. 212 List of Publications 213 [7] “A note on harmonic functions” (with Feng Li), Beijing Daxue Xuebao Ziran Kexue Ban, 43, no. 3, pp. 296–298, 2007. [8] “Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation” (with Weizhu Bao), Math. Comp., to appear. [9] “ Uniform error estimates of finite difference methods for the nonlinear Schr¨ odinger equation with wave operator” (with Weizhu Bao), SIAM J Numer. Anal., to appear. [10] “Gross-Pitaevskii-Poisson equation for dipolar Bose-Einstein condensate with anisotropic confinement” (with Weizhu Bao and Naoufel Ben Abdallah), preprint. [11] “Effective dipole-dipole interactions in multilayered dipolar Bose-Einstein condensates” (with Weizhu Bao and Matthias Rosenkranz), preprint. [12] “Breathing oscillations of a trapped impurity in a Bose gas” (with T. H. Johnson, M. Bruderer, S. R. Clark, W. Bao, and D. Jaksch), preprint. [...]... points For numerical comparisons between different numerical methods for GPE, or in a more general case, for the nonlinear Schr¨dinger equation (NLSE), we refer to [25, 47, 105, 144] o and references therein For ground states, along the theoretical front, Lieb et al [98] proved the existence and uniqueness of the positive ground state in three dimensions Along the numerical front, various numerical methods. .. this new formulation, analytical results on ground states and dynamics are presented Accurate and efficient numerical methods are proposed to compute the ground states and the dynamics Then, we derive the lower dimensional equations (one and two dimensions) for the three dimensional GPE (1.12) with anisotropic trapping potential Consequently, ground states and dynamics for the lower dimensional equations. .. O(ε2 ) and O(ε4 ) amplitudes for ill-prepared and well-prepared initial data, respectively This high oscillation in time brings significant difficulties in establishing error estimates uniformly in ε of the standard finite difference methods for NLSW, such as CNFD and SIFD Using new technical tools, we obtain error bounds uniformly in ε, at the order of O(h2 + τ 2/3 ) and O(h2 + τ ) with time step τ and mesh... time step τ and mesh size h for ill-prepared and well-prepared initial data, respectively, for both CNFD and SIFD in the l2 -norm and discrete semi-H 1 norm In addition, our error bounds are valid for general nonlinearity f (·) (1.16) in one, two and three dimensions In Chapter 8, we draw some conclusion and discuss some future work 10 Chapter 2 Gross- Pitaevskii equation for degenerate dipolar quantum... efficient and accurate numerical methods are designed for finding the ground states Chapter 6 is devoted to the numerical analysis for the finite difference discretizations applied to the rotational GPE ((1.12) with λ = 0), in two and three dimensions The optimal convergence rates are obtained for conservative Crank-Nicolson finite difference (CNFD) method and semi-implicit finite difference (SIFD) method for discretizing... two drawbacks and thus they are more accurate than those currently used in the literatures The key step is to decouple the dipolar interaction potential into a short-range and a long-range interaction (see (2.17) for details) and thus we can reformulate the GPE (2.5) into a GrossPitaevskii-Poisson type system In addition, based on the new mathematical formulation, we can prove existence and uniqueness... 2.3 A numerical method for computing ground states 26 Various numerical methods have been proposed in the literatures for computing the ground states of BEC (see [10, 15, 18, 39, 48, 50, 126] and references therein) One of the popular and efficient techniques for dealing with the constraint (2.11) is through the following construction [10, 12, 15]: Choose a time step ∆t > 0 and set tn = n ∆t for n =... study and structure of thesis f : [0, +∞) → R is a real-valued function Formally, when ε → 0+ , NLSW will converge to the standard NLSE [31, 129] We will investigate the impact of the parameter ε in the convergence rates for the finite difference discretizations of NLSW (1.16) 1.5 Purpose of study and structure of thesis This work is devoted to the mathematical analysis and numerical investigation for. .. second term in the Fourier transform of the dipolar interaction potential is 0 -type for 0-mode, i.e 0 when ξ = 0 (see (2.18) for details), and it is artificially omitted when ξ = 0 in practical computation [33, 70, 113, 122, 160, 163, 164] thus this may cause some numerical problems too The main aim of this chapter is to propose new numerical methods for computing ground states and dynamics of dipolar BECs... order O(h2 + τ 2 ) with time step τ and mesh size h, in both discrete l2 norm and discrete semi-H 1 norm Moreover, we make numerical 9 1.5 Purpose of study and structure of thesis comparison between CNFD and SIFD and conclude that SIFD is preferable in practical computation In Chapter 7, we investigate the uniform convergence rates (resp to ε) for finite difference methods applied to NLSW (1.16) The solution . MATHEMATICAL THEORY AND NUMERICAL METHODS FOR GROSS- P ITAEVSKII EQUATIONS AND APPLICATIONS CAI YONGYONG (M.Sc., Peking University) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR. physics, and motivates nu- merous mathematical and numerical studies on GPE. 1.1 The Gross- Pitaevskii equation Many different physical applications lead to the Gross- Pitaevskii equation (GPE). For example,. in proving uniform convergence rates f or CNFD and SIFD, independent of the perturbation. We overcome the difficulties and obtain uniform error bounds for both CNFD and SIFD, in one, two and three