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A quasi-local Gross-Pitaevskii equation for attractive Bose-Einstein condensates. (English) Zbl 1034.35126

Summary: We study a quasi-local approximation for a nonlocal nonlinear Schrödinger equation. The problem is closely related to several applications, in particular to Bose-Einstein condensates with attractive two-body interactions. The nonlocality is approximated by a nonlinear dispersion term, which is controlled by physically meaningful parameters. We show that the phenomenology found in the nonlocal model is very similar to that present in the reduced one with the nonlinear dispersion. We prove rigorously the absence of collapse in the model, and obtain numerically its stable soliton-like ground state.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
82B10 Quantum equilibrium statistical mechanics (general)
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[1] L. Vázquez, L. Streit, V.M. Pérez-Garcı́a (Eds.), Nonlinear Schrödinger and Klein-Gordon Systems: Theory and Applications, World Scientific, Singapore, 1996.
[2] Tod, P.; Moroz, I., An analytical approach to the schrödinger – newton equations, Nonlinearity, 12, 201-216, (1999) · Zbl 0942.35077
[3] Parola, A.; Salasnich, L.; Reatto, L., Structure and stability of bosonic clouds: alkali – metal atoms with negative scattering length, Phys. rev. A, 57, R3180-R3183, (1998)
[4] Pérez-Garcı́a, V.M.; Konotop, V.V.; Garcı́a-Ripoll, J.J., Dynamics of quasicollapse in nonlinear schroödinger systems with nonlocal interactions, Phys. rev. E, 62, 4300-4308, (2000)
[5] Goral, K.; Rzazewski, K.; Pfau, T., Bose – einstein condensation with magnetic dipole – dipole forces, Phys. rev. A, 61, 51601R-051604, (2000)
[6] O’Dell, D.S.; Giovanazzi, S.; Kurizki, G.; Akulin, V.M., Bose – einstein condensates with 1/r interatomic attraction: electromagnetically induced gravity, Phys. rev. lett., 84, 5687-5690, (2000)
[7] Brezzi, F.; Markowich, P.A., The three-dimensional wigner – poisson problem: existence, uniqueness and approximation, Math. mod. meth. appl. sci., 14, 35, (1991) · Zbl 0739.35080
[8] López, J.L.; Soler, J., Asymptotic behaviour to the 3D Schrödinger/hartree – poisson and wigner – poisson systems, Math. mod. meth. appl. sci., 10, 923-943, (2000) · Zbl 1012.81032
[9] Dalfovo, F.; Giorgini, S.; Pitaevski, L.P.; Stringari, S., Theory of bose – einstein condensation in trapped gases, Rev. mod. phys., 71, 463-521, (1999)
[10] C. Sulem, P. Sulem, The Nonlinear Schrödinger Equation, Springer, Berlin, 1999. · Zbl 0928.35157
[11] Pérez-Garcı́a, V.M.; Michinel, H.; Cirac, J.I.; Lewenstein, M.; Zoller, P., Dynamics of bose – einstein condensates: variational solutions of the gross – pitaevskii equations, Phys. rev. A, 56, 1424, (1997)
[12] Tsurumi, T.; Wadati, M., Stability of the D-dimensional nonlinear Schrödinger equation under confined potentials, J. phys. soc. jpn., 68, 1531-1536, (1999) · Zbl 0974.35117
[13] Pérez-Garcı́a, V.M.; Michinel, H.; Herrero, H., Bose – einstein solitons in highly asymmetric traps, Phys. rev. A, 57, 3837, (1998)
[14] Roberts, J.L.; Claussen, N.R.; Cornish, S.L.; Donley, E.A.; Cornell, E.A.; Wieman, C.E., Controlled collapse of a bose – einstein condensate, Phys. rev. lett., 86, 4211-4214, (2001)
[15] L.D. Landau, E.M. Lifshitz, Quantum Mechanics, Pergamon Press, New York, 1977.
[16] S.K. Turitsyn, Spatial dispersion of nonlinearity and stability of multidimensional solitons, Theor. Mat. Fiz. 64 (1985) 797-801.
[17] Krolikowski, W.; Bang, O., Solitons in nonlocal nonlinear media: exact solutions, Phys. rev. E, 63, 016610, (2000)
[18] Fibich, G.; Papanicolau, G., Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension, SIAM J. appl. math., 60, 183-240, (1999) · Zbl 1026.78013
[19] Mihalache, D.; Mazilu, D.; Crasovan, L.-C.; Malomed, B.A.; Lederer, F., Three-dimensional spinning solitons in the cubic – quintic nonlinear medium, Phys. rev. E, 61, 7142-7145, (2000)
[20] Vautherin, V.; Brink, C., Hartree – fock calculations with skyrme’s interaction. I. spherical nuclei, Phys. rev. C, 5, 626-647, (1972)
[21] Stringari, S.; Treiner, T., Surface properties of liquid 3he and 4he: a density-functional approach, Phys. rev. B, 36, 8369-8375, (1987)
[22] Davydova, T.A.; Fishchuk, A.I., Upper hybrid nonlinear wave structures, Ukr. J. phys., 40, 487, (1995)
[23] Garcı́a-Ripoll, J.J.; Pérez-Garcı́a, V.M., Optimizing Schrödinger functionals using Sobolev gradients, SIAM J. sci. comput., 23, 1215-1233, (2001)
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