A normalized gradient flow method with attractive-repulsive splitting for computing ground states of Bose-Einstein condensates with higher-order interaction

摘要

In this paper, we propose an efficient and stable method for computing groundstates of Bose-Einstein condensates (BEC) with higher order interaction (HOI),which can be described by the modified Gross-Pitaevskii equation (MGPE). Themethod is generalized from the popular normalized gradient flow method, whichhas been widely used for the classical GPE. The key point is to split the HOIterm into two parts, with the Hamiltonian corresponding to one term isattractive while the Hamiltonian corresponding to the other one is repulsive.Based on whether the corresponding Hamiltonian is attractive or repulsive, thetwo terms are treated in different ways in the numerical scheme. A variety ofnumerical experiments in 1D will be performed to show that the numericalmethods constructed in naive ways have high restrictions on time steps andtherefore inappropriate for the practical use, while the new method where theHOI term is splitted based on the attractive-repulsive splitting will overcomethe problem. Finally, we will show that our method can be applied to themultidimensional problems and to the computation of the first excited state aswell.