Please use this identifier to cite or link to this item:
|Title:||Mathematical theory and numerical methods for Bose-Einstein condensation|
|Authors:||Bao, W. |
|Citation:||Bao, W., Cai, Y. (2013). Mathematical theory and numerical methods for Bose-Einstein condensation. Kinetic and Related Models 6 (1) : 1-135. ScholarBank@NUS Repository. https://doi.org/10.3934/krm.2013.6.1|
|Abstract:||In this paper, we mainly review recent results on mathematical theory and numerical methods for Bose-Einstein condensation (BEC), based on the Gross-Pitaevskii equation (GPE). Starting from the simplest case with one-component BEC of the weakly interacting bosons, we study the reduction of GPE to lower dimensions, the ground states of BEC including the existence and uniqueness as well as nonexistence results, and the dynamics of GPE including dynamical laws, well-posedness of the Cauchy problem as well as the finite time blow-up. To compute the ground state, the gradient flow with discrete normalization (or imaginary time) method is reviewed and various full discretization methods are presented and compared. To simulate the dynamics, both finite difference methods and time splitting spectral methods are reviewed, and their error estimates are briefly outlined. When the GPE has symmetric properties, we show how to simplify the numerical methods. Then we compare two widely used scalings, i.e. physical scaling (commonly used) and semiclassical scaling, for BEC in strong repulsive interaction regime (Thomas-Fermi regime), and discuss semiclassical limits of the GPE. Extensions of these results for one-component BEC are then carried out for rotating BEC by GPE with an angular momentum rotation, dipolar BEC by GPE with long range dipole-dipole interaction, and two-component BEC by coupled GPEs. Finally, as a perspective, we show briefly the mathematical models for spin-1 BEC, Bogoliubov excitation and BEC at finite temperature. © American Institute of Mathematical Sciences.|
|Source Title:||Kinetic and Related Models|
|Appears in Collections:||Staff Publications|
Show full item record
Files in This Item:
There are no files associated with this item.
checked on May 21, 2018
WEB OF SCIENCETM
checked on Apr 25, 2018
checked on May 4, 2018
Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.