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East Asian Journal on Applied Mathematics, 1 (2011), pp. 49-81. |
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Ground States of Two-component Bose-Einstein Condensates with an Internal Atomic Josephson Junction Weizhu Bao 1*, Yongyong Cai 2 1 Department of Mathematics and Center for Computational Science and Engineering, National University of Singapore, Singapore 117543.2 Department of Mathematics, National University of Singapore, Singapore 117543. Received 19 March 2010; Accepted (in revised version) 17 May 2010 Available online 26 October 2010 doi:10.4208/eajam.190310.170510a Abstract In this paper, we prove existence and uniqueness results for the ground states of the coupled Gross-Pitaevskii equations for describing two-component Bose-Einstein condensates with an internal atomic Josephson junction, and obtain the limiting behavior of the ground states with large parameters. Efficient and accurate numerical methods based on continuous normalized gradient flow and gradient flow with discrete normalization are presented, for computing the ground states numerically. A modified backward Euler finite difference scheme is proposed to discretize the gradient flows. Numerical results are reported, to demonstrate the efficiency and accuracy of the numerical methods and show the rich phenomena of the ground sates in the problem. AMS subject classifications: 35Q55, 49J45, 65N06, 65N12, 65Z05, 81-08 Key words: Bose-Einstein condensate, coupled Gross-Pitaevskii equations, two-component, ground state, normalized gradient flow, internal atomic Josephson junction, energy *Corresponding author. Email: bao@math.nus.edu.sg (W. Bao), caiyongyong@nus.edu.sg (Y. Cai) |