https://doi.org/10.1140/epjb/e2010-00052-3
Stability and collapse of localized solutions of the controlled three-dimensional Gross-Pitaevskii equation
1
Dipartimento di Scienze Fisiche, Università Federico II and INFN Sezione di
Napoli, Complesso Universitario di M.S. Angelo, via Cintia, 80126 Napoli, Italy
2
Institute of Physics, P.O. Box 57, 11001 Belgrade, Serbia
3
Institut für Theoretische Physik IV, Ruhr-Universität Bochum, 44780 Bochum, Germany
4
Department of Physics, Umeå University, 90 187 Umeå, Sweden
5
Istituto di Cibernetica “Eduardo Caianiello” del CNR, Comprensorio “A. Olivetti” Fabbr. 70, Via Campi Flegrei, 34, 80078 Pozzuoli (NA), Italy
Corresponding authors: renato.fedele@na.infn.it djovanov@phy.bg.ac.yu bengt@tp4.rub.de s.denicola@cib.na.cnr.it ps@tp4.rub.de
Received:
7
August
2009
Revised:
10
January
2010
Published online:
16
February
2010
On the basis of recent investigations, a newly developed analytical procedure is used for constructing a wide class of localized solutions of the controlled three-dimensional (3D) Gross-Pitaevskii equation (GPE) that governs the dynamics of Bose-Einstein condensates (BECs) in the presence of a spatio-temporally varying external potential. The controlled 3D GPE is decomposed into a two-dimensional (2D) linear Schrödinger equation (called the 'transverse equation') and a one-dimensional (1D) nonlinear Schrödinger equation (called the 'longitudinal equation'), constrained by a variational condition for the controlling potential. The latter corresponds to the requirement for the minimization of the control operation in the transverse plane. Then, the above class of localized solutions are constructed as the product of the solutions of the transverse and longitudinal equations. A consistency condition between the transverse and longitudinal solutions yields a relationship between the transverse and longitudinal restoring forces produced by the external trapping potential well through a 'controlling parameter' (i.e. the average, with respect to the transverse profile, of the nonlinear inter-atomic interaction term of the GPE). It is found that the longitudinal profile supports localized solutions in the form of bright, dark or grey solitons with time-dependent amplitudes, widths and centroids. The related longitudinal phase is varying in space and time with time-dependent curvature radius and wavenumber. In turn, all the above parameters (i.e. amplitudes, widths, centroids, curvature radius and wavenumbers) can be easily expressed in terms of the controlling parameter. It is also found that the transverse profile has the form of Hermite-Gauss functions (depending on the transverse coordinates), and the explicit spatio-temporal dependence of the controlling potential is self-consistently determined. On the basis of these exact 3D analytical solutions, a stability analysis is carried out, focusing our attention on the physical conditions for having collapsing or non-collapsing solutions.
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2010
