Eur. Phys. J. B 54, 113-119 (2006)
https://doi.org/10.1140/epjb/e2006-00418-0

1D stability analysis of filtering and controlling the solitons in Bose-Einstein condensates

¹ Istituto di Cibernetica “Eduardo Caianiello” del CNR Comprensorio “A. Olivetti” Fabbr. 70, Via Campi Flegrei, 34, 80078 Pozzuoli (NA), Italy
² Dipartimento di Scienze Fisiche, Università Federico II and INFN Sezione di Napoli, Complesso Universitario di M.S. Angelo, via Cintia, 80126 Napoli, Italy
³ Institute of Physics, P.O. Box 57, 11001 Belgrade, Serbia
⁴ Department of Interdisciplinary Studies, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv, 69978, Israel
⁵ P.N. Lebedev Physical Institute, Leninskii Prospect 53, Moscow, 119991, Russia
⁶ Institut für Theoretische Physik IV and Centre for Plasma Science and Astrophysics, Fakultät für Physik und Astronomie, Ruhr–Universität Bochum, 44780 Bochum, Germany

Corresponding author: ^a renato.fedele@na.infn.it

Received: 28 August 2006
Published online: 24 November 2006

Abstract

We present one-dimensional (1D) stability analysis of a recently proposed method to filter and control localized states of the Bose–Einstein condensate (BEC), based on novel trapping techniques that allow one to conceive methods to select a particular BEC shape by controlling and manipulating the external potential well in the three-dimensional (3D) Gross–Pitaevskii equation (GPE). Within the framework of this method, under suitable conditions, the GPE can be exactly decomposed into a pair of coupled equations: a transverse two-dimensional (2D) linear Schrödinger equation and a one-dimensional (1D) longitudinal nonlinear Schrödinger equation (NLSE) with, in a general case, a time-dependent nonlinear coupling coefficient. We review the general idea how to filter and control localized solutions of the GPE. Then, the 1D longitudinal NLSE is numerically solved with suitable non-ideal controlling potentials that differ from the ideal one so as to introduce relatively small errors in the designed spatial profile. It is shown that a BEC with an asymmetric initial position in the confining potential exhibits breather-like oscillations in the longitudinal direction but, nevertheless, the BEC state remains confined within the potential well for a long time. In particular, while the condensate remains essentially stable, preserving its longitudinal soliton-like shape, only a small part is lost into “radiation”.