An explicit unconditionally stable numerical method for solving damped nonlinear Schrödinger equations with a focusing nonlinearity

SIAM Journal on Numerical Analysis

Volumn 41, Issue 4, 2003, Pages 1406-1426

  Bao, Weizhu ^a   Jaksch, Dieter ^b  

^a NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

^b UNIVERSITY OF INNSBRUCK   (Austria)

Author keywords

Bose Einstein condensate (BEC); Complex Ginzburg Landau (CGL); Damped nonlinear Schr dinger equation (DNLS); Gross Pitaevskii equation (GPE); Time splitting sine spectral (TSSP) method

Indexed keywords

COLLISION AVOIDANCE; DAMPING; NONLINEAR EQUATIONS; QUANTUM THEORY; SPECTRUM ANALYSIS; WAVE EQUATIONS;

BOSE-EINSTEIN CONDENSATE (BEC); COMPLEX GINZBURG-LANDAU (CGL); DAMPED NONLINEAR SCHRODINGER EQUATION (DNLS); GROSS-PITAEVSKII EQUATION (GPE); TIME-SPLITTING SINE-SPECTRAL (TSSP) METHOD;

NUMERICAL METHODS;

EID: 0442300961     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142902413391     Document Type: Article

References (40)

1. S. K. ADHIKARI, Mean-field description for collapsing and exploding Bose-Einstein condensates, Phys. Rev. A, 66 (2002), pp. 3611-3615.
2. 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.
3. J. R. ANGLIN AND W. KETTERLE, Bose-Einstein condensation of atomic gases, Nature, 416 (2002), pp. 211-218.
4. W. BAO, S. JIN, AND P. A. MARKOWICH, On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comput. Phys., 175 (2002), pp. 487-524.
5. 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), pp. 27-64.
6. W. BAO AND W. TANG, Ground state solution of trapped interacting Bose-Einstein condensate by minimizing a functional, J. Comput. Phys., 187 (2003), pp. 230-254.
7. W. BAO, D. JAKSCH, AND P. A. MARKOWICH, Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation, J. Comput. Phys., 187 (2003), pp. 318-342.
8. W. BAO, D. JAKSCH. AND P. A. MARKOWICH, Three Dimensional Simulation of Jet Formation in Collapsing Condensates, preprint, 2003. Available online from http://arXiv.org/abs/cond-mat/0307344.
9. R. CIEGIS AND V. PAKALNYTE, The finite difference scheme for the solution of weakly damped nonlinear Schrödinger equation, Internat. J. Appl. Sci. Comput., 8 (2001), pp. 127-134.
10. E. CORNELL, Very cold indeed: The nanokelvin physics of Bose-Einstein condensation, J. Res. Natl. Inst. Stan., 101 (1996), pp. 419-434.
11. F. DALFOVO, S. GIORGINI, L. P. PITAEVSKII, AND S. STRINGARI, Theory of Bose-Einstein condensation in trapped gases, Rev. Modern Phys., 71 (1999), pp. 463-512.
12. E. A. DONLEY, N. R. CLAUSSEN, S. L. CORNISH, J. L. ROBERTS, E. A. CORNELL, AND C. E. WIEMAN, Dynamics of collapsing and exploding Bose-Einstein condensates, Nature, 412 (2001), pp. 295-299.
13. R. A. DUINE AND H. T. C. STOOF, Explosion of a collapsing Bose-Einstein condensate, Phys. Rev. Lett., 86 (2001), pp. 2204-2207.
14. G. FIBICH, Self-focusing in the damped nonlinear Schrödinger equation, SIAM J. Appl. Math., 61 (2001), pp. 1680-1705.
15. G. FIBICH AND D. LEVY, Self-focusing in the complex Ginzburg-Landau limit of the critical nonlinear Schrödinger equation, Phys. Lett. A, 249 (1998), pp. 286-294.
16. G. FIBICH AND G. PAPANICOLAOU, Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension, SIAM J. Appl. Math., 60 (1999), pp. 183-240.
17. 2, J. Differential Equations, 165 (2000), pp. 96-122.
18. 22, Adv. Differential Equations, 3 (1998), pp. 337-360.
19. O. GOUBET, Approximate inertial manifolds for a weakly damped nonlinear Schrödinger equation, Discrete Contin. Dynam. Systems, 3 (1997), pp. 503-530.
20. M. GREINER, O. MANDEL, T. ESSLINGER, T. W. HÄNSCH, AND I. BLOCH, Quantum phase transition from a superfluid to a mott insulator in a gas of ultracold atoms, Nature, 415 (2002), pp. 39-45.
21. E. P. GROSS, Structure of a quantized vortex in boson systems, Nuovo Cimento (10), 20 (1961), pp. 454-477.
22. S. R. K. IYENGAR, G. JAYARAMAN, AND V. BALASUBRAMANIAN, Variable mesh difference schemes for solving a nonlinear Schrödinger equation with a linear damping term. Advances in partial differential equations, III, Comput. Math. Appl., 40 (2000), pp. 1375-1385.
23. D. JAKSCH, C. BRUDER, J. I. CIRAC, C. W. GARDINER, AND P. ZOLLER, Cold bosonic atoms in optical lattices, Phys. Rev. Lett., 81 (1998), pp. 3108-3111.
24. M. S. JOLLY, R. TEMAM, AND C. XIONG, An application of approximate inertial manifolds to a weakly damped nonlinear Schrödinger equation, Numer. Funct. Anal. Optim., 16 (1995), pp. 923-937.
25. Y. KAGAN, A. E. MURYSHEV, AND G. V. SHLYAPNIKOV, Collapse and Bose-Einstein condensation in a trapped Bose gas with negative scattering length, Phys. Rev. Lett., 81 (1998), pp. 933-937.
26. L. LANDAU AND E. LIFSCHITZ, Quantum Mechanics: Non-Relativistic Theory, Pergamon Press, New York, 1977.
27. M. J. LANDMAN, G. C. PAPANICOLAOU, C. SULEM, P. L. SULEM, AND X. P. WANG, Stability of isotropic singularities for the nonlinear Schrödinger equation, Phys. D, 47 (1991), pp. 393-415.
28. A. MIELKE, The complex Ginzburg-Landau equation on large and unbounded domains: Sharper bounds and attractors, Nonlinearity, 10 (1997), pp. 199-222.
29. G. MOEBS, Guy A multilevel method for the resolution of a stochastic weakly damped nonlinear Schrödinger equation, Appl. Numer. Math., 26 (1998), pp. 353-375.
30. G. MOEBS AND R. TEMAM, Resolution of a stochastic weakly damped nonlinear Schrödinger equation by a multilevel numerical method, J. Opt. Soc. Amer. A, 17 (2000), pp. 1870-1879.
31. G. C. PAPANICOLAOU, C. SULEM, P. L. SULEM, AND X. P. WANG, Singular solutions of the Zakharov equations for Langmuir turbulence, Phys. Fluids B, 3 (1991), pp. 969-980.
32. L. S. PERANICH, A finite difference scheme for solving a non-linear Schrödinger equation with a linear damping term, J. Comput. Phys., 68 (1987), pp. 501-505.
33. L. P. PITAEVSKII, Vortex lines in an imperfect Bose gaze, Z. Èksper. Teoret. Fiz., 40 (1961), pp. 646-651 (in Russian); Soviet Physics JETP, 13 (1961), pp. 451-454.
34. L. P. PITAEVSKII, Vortex lines in an imperfect Bose gaze, Z. Èksper. Teoret. Fiz., 40 (1961), pp. 646-651 (in Russian); Soviet Physics JETP, 13 (1961), pp. 451-454.
35. 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), pp. 728-731.
36. H. SAITO AND M. UEDA, Intermittent implosion and pattern formation of trapped Bose-Einstein condensates with an attractive interaction, Phys. Rev. Lett., 86 (2001), pp. 1406-1409.
37. C. SULEM AND P. L. SULEM, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, Springer-Verlag, New York, 1999.
38. M. TSUTSUMI, Nonexistence of global solutions to the Cauchy problem for the damped nonlinear Schrödinger equations, SIAM J. Math. Anal., 15 (1984), pp. 357-366.
39. M. TSUTSUMI, On global solutions to the initial-boundary value problem for the damped nonlinear Schrödinger equations, J. Math. Anal. Appl., 145 (1990), pp. 328-341.
40. F.-Y. ZHANG AND S.-J. LU, Long-time behavior of finite difference solutions of a nonlinear Schrödinger equation with weakly damped, J. Comput. Math., 19 (2001), pp. 393-406.