Title: A normalized gradient flow method with attractive-repulsive splitting for computing ground states of Bose-Einstein condensates with higher-order interaction,
Abstract: In this paper, we generalize the normalized gradient flow method to compute
the ground states of Bose-Einstein condensates (BEC) with higher order
interactions (HOI), which is modelled via the modified Gross-Pitaevskii
equation (MGPE). Schemes constructed in naive ways suffer from severe stability
problems due to the high restrictions on time steps. To build an efficient and
stable scheme, we split the HOI term into two parts with each part treated
separately. The part corresponding to a repulsive/positive energy is treated
semi-implicitly while the one corresponding to an attractive/negative energy is
treated fully explicitly. Based on the splitting, we construct the
BEFD-splitting and BESP-splitting schemes. A variety of numerical experiments
shows that the splitting will improve the stability of the schemes
significantly. Besides, we will show that the methods can be applied to
multidimensional problems and to the computation of the first excited state as
well.