Exploring bifurcations in Bose–Einstein condensates via phase field crystal models

To facilitate the analysis of pattern formation and the related phase transitions in Bose–Einstein condensates, we present an explicit approximate mapping from the nonlocal Gross–Pitaevskii equation with cubic nonlinearity to a phase field crystal (PFC) model. This approximation is valid close to the superfluid–supersolid phase transition boundary. The simplified PFC model permits the exploration of bifurcations and phase transitions via numerical path continuation employing standard software. While revealing the detailed structure of the bifurcations present in the system, we demonstrate the existence of localized states in the PFC approximation. Finally, we discuss how higher-order nonlinearities change the structure of the bifurcation diagram representing the transitions found in the system.