Scientific Results

Mathematical and physical aspects of controlling the exact solutions of the 3D Gross-Pitaevskii equation

Year: 2010

Authors: Fedele R., Jovanovic D., De Nicola S., Eliasson B., Shukla P.K.

Autors Affiliation: Univ Naples Federico 2, Dipartimento Sci Fis, I-80126 Naples, Italy; Univ Strathclyde, Dept Phys, SUPA, Glasgow G4 ONG, Lanark, Scotland; Umea Univ, Dept Phys, SE-90187 Umea, Sweden; Ruhr Univ Bochum, Inst Theoret Phys 4, D-44780 Bochum, Germany; Inst Phys, Belgrade 11001, Serbia; CNR, Ist Cibernet Eduardo Caianiello, I-80078 Pozzuoli, NA, Italy; Ist Nazl Fis Nucl, Sez Napoli, I-80126 Naples, Italy

Abstract: The possibility of the decomposition of the three-dimensional (3D) Gross-Pitaevskii equation (GPE) into a pair of coupled Schrodinger-type equations, is investigated. It is shown that, under suitable mathematical conditions, it is possible to construct the exact controlled solutions of the 3D GPE from the solutions of a linear 2D Schrodinger equation coupled with a I D nonlinear Schrodinger equation (the transverse and longitudinal components of the GPE, respectively). The coupling between these two equations is the functional of the transverse and the longitudinal profiles. The applied method of nonlinear decomposition, called the controlling potential method (CPM), yields the full 3D solution in the form of the product of the solutions of the transverse and longitudinal components of the GPE. It is shown that the CPM constitutes a variational principle and sets up a condition on the controlling potential well. Its physical interpretation is given in terms of the minimization of the (energy) effects introduced by the control. The method is applied to the case of a parabolic external potential to construct analytically an exact BEC state in the form of a bright soliton, for which the quantitative comparison between the external and controlling potentials is presented. (C) 2009 Elsevier B.V. All rights reserved.


Volume: 374 (5)      Pages from: 788  to: 795

More Information: This work was partially supported by Fondo Affari Internazionali of Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, (Napoli, Italy), by the Deutsche Forschungsgemeinschaft (Bonn, Germany) through the project SH21/3-1 of the Research Unit 1048, by the Serbian MNTR grant 141031, and by the Swedish Research Council (VR).
DOI: 10.1016/j.physleta.2009.11.069

Citations: 7
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