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Title: Dimension reduction of the Gross-Pitaevskii equation for Bose-Einstein condensates
Authors: GE YUNYI
Keywords: Gross-Pitaevskii equation, Bose-Einstein condensate, Normalized gradient flow, Ground state solution, Dynamics, Dimension reduction
Issue Date: 29-Mar-2005
Citation: GE YUNYI (2005-03-29). Dimension reduction of the Gross-Pitaevskii equation for Bose-Einstein condensates. ScholarBank@NUS Repository.
Abstract: We study numerically and asymptotically dimension reduction of 3D GPE forBEC in certain limiting trapping frequency regimes. First, we take the 3D GPE,scale it to get a three parameters model, and review how to reduce it to 2D GPEin disk-shaped condensation or 1D GPE in cigar-shaped condensation. Then wecompute the ground state of 3D GPE numerically by a normalized gradient flow un-der backward Euler finite difference discretization and verify numerically the formaldimension reduction for ground state. Furthermore, we obtain Thomas-Fermi andfirst order approximations for energy and chemical potential of the ground state ford-dimension GPE with d = 1; 2; 3. Then we classify approximations of the groundstate of 3D GPE in three cases based on the ratios between the trapping frequen-cies: i). isotropic condensation; ii). disk-shaped condensation; iii). cigar-shapedcondensation. These results are fully confirmed by our 3D numerical results. Also,convergence rates in relative error for all interacting quantities are observed andreported. Finally, we study dimension reduction of time-dependent GPE from 3Dto 2D numerically by a fourth-order time-splitting sine-spectral method. Our nu-merical results confirm the formal dimension reduction for time-dependent GPE andalso suggest convergence rates in limiting trapping frequency ratios.
Appears in Collections:Master's Theses (Open)

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