Abstract

The Gross-Pitaevskii equation (GPE) has proven to be a big success in describing the Bose-Einstein condensate (BEC), where the interaction between particles is approximated by the binary contact interaction. However, in the case of higher particle densities, some correction terms need to be included for a better description. And one choice is to include a higher order interaction (HOI) term in the binary interaction part, which will lead to the modified GPE (MGPE).

In the talk, I will study the ground states of the MGPE analytically, asymptotically and numerically. I will start by introducing the model, focusing on the dimension reduction problem. And then I will show the existence and uniqueness of ground states, together with a detailed characterization of the Thomas-Fermi (TF) approximations under two special external potentials, i.e. the harmonic potential and the box potential. Finally, I will introduce two numerical methods for computing the ground states. Both algorithms are well-adapted to deal with the HOI term.