Multiple peak solutions for Bose-Einstein condensates in a periodic potential

We study multiple peak solutions of Bose-Einstein condensates (BEC) in a periodic potential. A staircase multigrid-continuation algorithm is presented for computing energy levels of the BEC. The proposed algorithm is a modification of the two-grid discretization schemes [6] or the simplified two-grid schemes [22] for tracing solution curves of semilinear elliptic eigenvalue problems. The algorithm has the following advantages over the two-grid or the simplified two-grid schemes: (i) It guarantees that the scheme will converge to the target point on the finest grid. (ii) It is cheaper than the simplified two-grid scheme. We apply the staircase-multigrid continuation algorithm to study the ground-state and the firstfew excited solutions of the 1D, 2D and 3D BEC in a periodic potential. Our numerical results show that if the chemical potential is large enough, the number of peaks for the ground-state solutions is n ∏ j=1 ( 1 dj − 1), where d1 is the distance of neighbor wells in the xcoordinate, and so on, and n the dimension of the BEC. Moreover, the number of peaks is less than n ∏ j=1 ( 1 dj −1) for the excited-state solutions. Additionally, the global wave number of the ground-state and the excited-state solutions is the same as that of the associated linear eigenvalue problem if di is/are small enough. The numerical results are consistent with the mathematical formulation and the theoretical prediction of the BEC in a periodic potential. ∗Corresponding author. Supported by the National Science Council of R.O.C. (Taiwan) through Project NSC 95-2115-M-005-004-MY3. E-mail: [email protected] (S.-L. Chang), [email protected] (H.-S. Chen), [email protected] (C.-S. Chien).