  To facilitate the analysis of pattern formation and of the related phase
transitions in Bose-Einstein condensates (BECs) 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. Finally, we discuss how higher-order
nonlinearities change the structure of the bifurcation diagram representing the
transitions found in the system.
