The inverse problem for the Gross-Pitaevskii equation

Malomed, Boris A. and Stepanyants, Yury A. (2010) The inverse problem for the Gross-Pitaevskii equation. Chaos: an interdisciplinary journal of nonlinear science, 20 (1). p. 13130. ISSN 1054-1500

Abstract

Two different methods are proposed for the generation of wide classes of exact solutions to the stationary Gross-Pitaevskii equation (GPE). The first method, suggested by the work by Kondrat'ev and Miller (1966), applies to one-dimensional GPE. It is based on the similarity between the GPE and the integrable Gardner equation, all solutions of the latter equation (both stationary and nonstationary ones) generating exact solutions to the GPE. The second method is based on the 'inverse problem' for the GPE, i.e. construction of a potential function which provides a desirable solution to the equation. Systematic results are presented for one- and two-dimensional cases. Both methods are illustrated by a variety of localized solutions, including solitary vortices, for both attractive and repulsive nonlinearity in the GPE. The stability of the one-dimensional solutions is tested by direct simulations of the time-dependent GPE.


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Item Type: Article (Commonwealth Reporting Category C)
Refereed: Yes
Item Status: Live Archive
Additional Information: © 2010 American Institute of Physics. Permanent restricted access to published version due to publisher copyright policy.
Depositing User: Assoc Prof Yury Stepanyants
Faculty / Department / School: Historic - Faculty of Sciences - Department of Maths and Computing
Date Deposited: 16 Apr 2010 06:39
Last Modified: 08 Dec 2014 23:58
Uncontrolled Keywords: Bose-Einstein condensation; Gross-Pitaevskii equation; localised stationary solutions; numerical method; nonlinear Schrodinger equation; inverse problem
Fields of Research (FOR2008): 01 Mathematical Sciences > 0105 Mathematical Physics > 010599 Mathematical Physics not elsewhere classified
01 Mathematical Sciences > 0105 Mathematical Physics > 010502 Integrable Systems (Classical and Quantum)
02 Physical Sciences > 0204 Condensed Matter Physics > 020403 Condensed Matter Modelling and Density Functional Theory
Socio-Economic Objective (SEO2008): E Expanding Knowledge > 97 Expanding Knowledge > 970102 Expanding Knowledge in the Physical Sciences
E Expanding Knowledge > 97 Expanding Knowledge > 970101 Expanding Knowledge in the Mathematical Sciences
Identification Number or DOI: 10.1063/1.3367776
URI: http://eprints.usq.edu.au/id/eprint/7314

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