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Dynamics of Bose-Einstein condensates: exact representation and topological classification of coherent matter waves. (English) Zbl 1470.82017

Summary: By using the bifurcation theory of dynamical systems, we present the exact representation and topological classification of coherent matter waves in Bose-Einstein condensates (BECs), such as solitary waves and modulate amplitude waves (MAWs). The existence and multiplicity of such waves are determined by the parameter regions selected. The results show that the characteristic of coherent matter waves can be determined by the “angular momentum” in attractive BECs while for repulsive BECs; the waves of the coherent form are all MAWs. All exact explicit parametric representations of the above waves are exhibited and numerical simulations support the result.

MSC:

82C10 Quantum dynamics and nonequilibrium statistical mechanics (general)
35Q55 NLS equations (nonlinear Schrödinger equations)
81P40 Quantum coherence, entanglement, quantum correlations
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[1] Davis, K. B.; Mewes, M.-O.; Andrews, M. R.; Van Druten, N. J.; Durfee, D. S.; Kurn, D. M.; Ketterle, W., Bose-Einstein condensation in a gas of sodium atoms, Physical Review Letters, 75, 22, 3969-3973 (1995)
[2] Burger, S.; Cataliotti, F. S.; Fort, C.; Minardi, F.; Inguscio, M.; Chiofalo, M. L.; Tosi, M. P., Superfluid and dissipative dynamics of a Bose-Einstein condensate in a periodic optical potential, Physical Review Letters, 86, 20, 4447-4450 (2001)
[3] Wu, B.; Diener, R. B.; Niu, Q., Bloch waves and bloch bands of Bose-Einstein condensates in optical lattices, Physical Review A, 65, 2 (2002)
[4] Chen, A.; Wen, S.; Huang, W., Existence and orbital stability of periodic wave solutions for the nonlinear Schrödinger equation, The Journal of Applied Analysis and Computation, 2, 2, 137-148 (2012) · Zbl 1304.35633
[5] Deconinck, B.; Frigyik, B. A.; Kutz, J. N., Dynamics and stability of Bose-Einstein condensates: the nonlinear Schrödinger equation with periodic potential, Journal of Nonlinear Science, 12, 3, 169-205 (2002) · Zbl 1009.35078
[6] Chong, G.; Hai, W.; Xie, Q., Spatial chaos of trapped Bose-Einstein condensate in one-dimensional weak optical lattice potential, Chaos, 14, 2, 217-223 (2004)
[7] Chong, G.; Hai, W.; Xie, Q., Controlling chaos in a weakly coupled array of Bose-Einstein condensates, Physical Review E, 71, 1 (2005)
[8] Dyke, P.; Lei, S.; Hulet, R., Phase-dependent Interactions of Bright Matter-Wave Solitons, Bulletin of the American Physical Society, 57, 5 (2012)
[9] Xu, Y.; Zhang, Y.; Wu, B., Bright solitons in spin-orbit-coupled Bose-Einstein condensates, Physical Review A, 87, 1 (2013)
[10] Schlein, B.; Yau, H. T., Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate, Annals of Mathematics, 172, 1, 291-370 (2010) · Zbl 1204.82028
[11] Krauth, W., Quantum Monte Carlo calculations for a large number of bosons in a harmonic trap, Physical Review Letters, 77, 18, 3695-3699 (1996)
[12] Nascimbène, S.; Chen, Y.; Atala, M.; Aidelsburger, M.; Trotzky, S.; Paredes, B.; Bloch, I., Experimental realization of plaquette resonating valence-bond states with ultracold atoms in optical superlattices, Physical Review Letters, 108, 20 (2012)
[13] Chua, V. P.; Porter, M. A., Spatial resonance overlap in Bose-Einstein condensates in superlattice potentials, International Journal of Bifurcation and Chaos, 16, 4, 945-959 (2006) · Zbl 1115.82304
[14] Porter, M. A.; Kevrekidis, P. G., Bose-Einstein condensates in super-lattices, SIAM Journal on Applied Dynamical Systems, 4, 4, 783-807 (2005) · Zbl 1145.82309
[15] Liu, Q.; Qian, D., Modulated amplitude waves with nonzero phases in Bose-Einstein condensates, Journal of Mathematical Physics, 52, 8 (2011) · Zbl 1272.82025
[16] Liu, Q.; Qian, D., Construction of modulated amplitude waves via averaging in collisionally inhomogeneous Bose-Einstein condensates, Journal of Nonlinear Mathematical Physics, 19, 2 (2012) · Zbl 1254.35209
[17] Porter, M. A.; Kevrekidis, P. G.; Malomed, B. A., Resonant and non-resonant modulated amplitude waves for binary Bose-Einstein condensates in optical lattices, Physica D, 196, 1-2, 106-123 (2004) · Zbl 1098.81881
[18] Li, X.; Han, J.; Wang, F., The extended Riccati equation method for travelling wave solutions of ZK equation, The Journal of Applied Analysis and Computation, 2, 4, 423-430 (2012) · Zbl 1317.74048
[19] Chow, S.; Jiang, M.; Lin, X., Traveling wave solutions in coupled Chua’s circuits, part I: periodic solutions, The Journal of Applied Analysis and Computation, 3, 3, 213-237 (2013) · Zbl 1307.34025
[20] Li, J.; Liu, Z., Traveling wave solutions for a class of nonlinear dispersive equations, Chinese Annals of Mathematics. Series B, 23, 3, 397-418 (2002) · Zbl 1011.35014
[21] Li, J.; Chen, G., On a class of singular nonlinear traveling wave equations, International Journal of Bifurcation and Chaos, 17, 11, 4049-4065 (2007) · Zbl 1158.35080
[22] Brusch, L.; Torcini, A.; van Hecke, M.; Zimmermann, M. G.; Bär, M., Modulated amplitude waves and defect formation in the one-dimensional complex Ginzburg-Landau equation, Physica D, 160, 3-4, 127-148 (2001) · Zbl 0996.35071
[23] Bronski, J. C.; Carr, L. D.; Deconinck, B.; Kutz, J. N., Bose-Einstein condensates in standing waves: the cubic nonlinear Schrödinger equation with a periodic potential, Physical Review Letters, 86, 8, 1402-1405 (2001)
[24] Byrd, P. F.; Friedman, M. D., Handbook of Elliptic Integrals for Engineers and Scientists. Handbook of Elliptic Integrals for Engineers and Scientists, Die Grundlehren der mathematischen Wissenschaften, 67, xvi+358 (1971), New York, NY, USA: Springer, New York, NY, USA
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