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Mathematical and physical aspects of controlling the exact solutions of the 3D Gross-Pitaevskii equation
Umeå University, Faculty of Science and Technology, Department of Physics.
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2010 (English)In: Physics Letters A, ISSN 0375-9601, Vol. 374, no 5, 788-795 p.Article in journal (Refereed) Published
Abstract [en]

The possibility of the decomposition of the three-dimensional (3D) Gross–Pitaevskii equation (GPE) into a pair of coupled Schrödinger-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 Schrödinger equation coupled with a 1D nonlinear Schrödinger 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.

Place, publisher, year, edition, pages
2010. Vol. 374, no 5, 788-795 p.
URN: urn:nbn:se:umu:diva-30623DOI: 10.1016/j.physleta.2009.11.069ISI: 000274607300017OAI: diva2:284998
Available from: 2010-01-10 Created: 2010-01-10 Last updated: 2011-01-26Bibliographically approved

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Eliasson, Bengt
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