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International Journal of Differential Equations
Volume 2013 (2013), Article ID 874196, 8 pages
http://dx.doi.org/10.1155/2013/874196
Research Article

Dynamics of a Gross-Pitaevskii Equation with Phenomenological Damping

Departamento de Matemáticas, Pontificia Universidad Javeriana, Carrera 7 No. 40-62, Bogotá, Colombia

Received 19 April 2013; Accepted 3 June 2013

Academic Editor: Norio Yoshida

Copyright © 2013 Renato Colucci et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. J. Bourgain, Global Solutions of Nonlinear Schrödinger Equations, vol. 46 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, USA, 1999. View at MathSciNet
  2. R. Carles, Semi-Classical Analysis for Nonlinear Schrödinger Equations, World Scientific, Hackensack, NJ, USA, 2008.
  3. T. Tao, Nonlinear Dispersive Equations, vol. 106 of CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC, USA, 2006. View at MathSciNet
  4. F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Universitext, Springer, New York, NY, USA, 2009. View at MathSciNet
  5. A. Aftalion, Vortices in Bose-Einstein Condensates, vol. 67 of Progress in Nonlinear Differential Equations and their Applications, Birkhäuser, Boston, Mass, USA, 2006. View at Zentralblatt MATH · View at MathSciNet
  6. W. Bao and Y. Cai, “Mathematical theory and numerical methods for Bose-Einstein condensation,” Kinetic and Related Models, vol. 6, no. 1, pp. 1–135, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  7. W. Bao, Q. Du, and Y. Zhang, “Dynamics of rotating Bose-Einstein condensates and its efficient and accurate numerical computation,” SIAM Journal on Applied Mathematics, vol. 66, no. 3, pp. 758–786, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. C. Hao, L. Hsiao, and H.-L. Li, “Global well posedness for the Gross-Pitaevskii equation with an angular momentum rotational term,” Mathematical Methods in the Applied Sciences, vol. 31, no. 6, pp. 655–664, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. 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,” Journal of Mathematical Physics, vol. 48, no. 10, Article ID 102105, 11 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. P. Antonelli, D. Marahrens, and C. Sparber, “On the Cauchy problem for nonlinear Schrödinger equations with rotation,” Discrete and Continuous Dynamical Systems A, vol. 32, no. 3, pp. 703–715, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. H. Liu, “Critical thresholds in the semiclassical limit of 2-D rotational Schrödinger equations,” Zeitschrift für Angewandte Mathematik und Physik, vol. 57, no. 1, pp. 42–58, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  12. S. Choi, S. A. Morgan, and K. Burnett, “Phenomenological damping in trapped atomic Bose-Einstein condensates,” Physical Review A, vol. 57, no. 5, pp. 4057–4060, 1998. View at Scopus
  13. P. G. Kevrekidis and D. J. Frantzeskakis, “Multiple dark solitons in Bose-Einstein condensates at finite temperatures,” Discrete and Continuous Dynamical Systems S, vol. 4, no. 5, pp. 1199–1212, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. M. Tsubota, K. Kasamatsu, and M. Ueda, “Vortex lattice formation in a rotating Bose-Einstein condensate,” Physical Review A, vol. 65, no. 2, Article ID 023603, 2002. View at Scopus
  15. K. Kasamatsu, M. MacHida, N. Sasa, and M. Tsubota, “Three-dimensional dynamics of vortex-lattice formation in Bose-Einstein condensates,” Physical Review A, vol. 71, no. 6, Article ID 063616, 2005. View at Publisher · View at Google Scholar · View at Scopus
  16. L. H. Wen and X. B. Luo, “Formation and structure of vortex lattices in a rotating double-well Bose-Einstein condensate,” Laser Physics Letters, vol. 9, no. 8, pp. 618–624, 2012.
  17. C. W. Gardiner, J. R. Anglin, and T. I. A. Fudge, “The stochastic Gross-Pitaevskii equation,” Journal of Physics B, vol. 35, no. 6, pp. 1555–1582, 2002. View at Publisher · View at Google Scholar · View at Scopus
  18. C. W. Gardiner and M. J. Davis, “The stochastic Gross-Pitaevskii equation: II,” Journal of Physics B, vol. 36, no. 23, pp. 4731–4753, 2003. View at Publisher · View at Google Scholar · View at Scopus
  19. A. S. Bradley and C. W. Gardiner, “The stochastic Gross-Pitaevskii equation III,” http://arxiv.org/abs/cond-mat/0602162.
  20. M. Kurzke, C. Melcher, R. Moser, and D. Spirn, “Dynamics for Ginzburg-Landau vortices under a mixed flow,” Indiana University Mathematics Journal, vol. 58, no. 6, pp. 2597–2622, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. E. Miot, “Damped wave dynamics for a complex Ginzburg-Landau equation with low dissipation,” http://128.84.158.119/abs/1003.5375v1.
  22. J.-M. Ghidaglia and B. Héron, “Dimension of the attractors associated to the Ginzburg-Landau partial differential equation,” Physica D, vol. 28, no. 3, pp. 282–304, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. P. Laurençot, “Long-time behaviour for weakly damped driven nonlinear Schrödinger equations in N, N3,” NoDEA. Nonlinear Differential Equations and Applications, vol. 2, no. 3, pp. 357–369, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. N. Kita and A. Shimomura, “Large time behavior of solutions to Schrödinger equations with a dissipative nonlinearity for arbitrarily large initial data,” Journal of the Mathematical Society of Japan, vol. 61, no. 1, pp. 39–64, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. P. Antonelli and C. Sparber, “Global well-posedness for cubic NLS with nonlinear damping,” Communications in Partial Differential Equations, vol. 35, no. 12, pp. 2310–2328, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. W. Bao, D. Jaksch, and P. A. Markowich, “Three-dimensional simulation of jet formation in collapsing condensates,” Journal of Physics B, vol. 37, no. 2, pp. 329–343, 2004. View at Publisher · View at Google Scholar · View at Scopus
  27. W. Bao and D. Jaksch, “An explicit unconditionally stable numerical method for solving damped nonlinear Schrödinger equations with a focusing nonlinearity,” SIAM Journal on Numerical Analysis, vol. 41, no. 4, pp. 1406–1426, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, vol. 68 of Applied Mathematical Sciences, Springer, New York, NY, USA, 2nd edition, 1997. View at MathSciNet
  29. C. Foiaş and G. Prodi, “Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension 2,” Rendiconti del Seminario Matematico della Università di Padova, vol. 39, pp. 1–34, 1967. View at MathSciNet
  30. J. C. Robinson, Infinite-Dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, Mass, USA, 2001. View at Publisher · View at Google Scholar · View at MathSciNet