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International Journal of Differential Equations
Volume 2013 (2013), Article ID 874196, 8 pages
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.
- J. Bourgain, Global Solutions of Nonlinear Schrödinger Equations, vol. 46 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, USA, 1999.
- R. Carles, Semi-Classical Analysis for Nonlinear Schrödinger Equations, World Scientific, Hackensack, NJ, USA, 2008.
- 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.
- F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Universitext, Springer, New York, NY, USA, 2009.
- A. Aftalion, Vortices in Bose-Einstein Condensates, vol. 67 of Progress in Nonlinear Differential Equations and their Applications, Birkhäuser, Boston, Mass, USA, 2006.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- A. S. Bradley and C. W. Gardiner, “The stochastic Gross-Pitaevskii equation III,” http://arxiv.org/abs/cond-mat/0602162.
- 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.
- E. Miot, “Damped wave dynamics for a complex Ginzburg-Landau equation with low dissipation,” http://184.108.40.206/abs/1003.5375v1.
- 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.
- P. Laurençot, “Long-time behaviour for weakly damped driven nonlinear Schrödinger equations in , ,” NoDEA. Nonlinear Differential Equations and Applications, vol. 2, no. 3, pp. 357–369, 1995.
- 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.
- 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.
- 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.
- 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.
- R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, vol. 68 of Applied Mathematical Sciences, Springer, New York, NY, USA, 2nd edition, 1997.
- 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.
- 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.