Stability and phase transition of localized modes in Bose-Einstein condensates with both two- and three-body interactions

We investigate the stability and phase transition of localized modes in Bose-Einstein Condensates (BECs) in an optical lattice with the discrete nonlinear Schr\"{o}dinger model by considering both two- and three-body interactions. We find that there are three types of localized modes, bright discrete breather (DB), discrete kink (DK), and multi-breather (MUB). Moreover, both two- and three-body on-site repulsive interactions can stabilize DB, while on-site attractive three-body interactions destabilize it. There is a critical value for the three-body interaction with which both DK and MUB become the most stable ones. We give analytically the energy thresholds for the destabilization of localized states and find that they are unstable (stable) when the total energy of the system is higher (lower) than the thresholds. The stability and dynamics characters of DB and MUB are general for extended lattice systems. Our result is useful for the blocking, filtering, and transfer of the norm in nonlinear lattices for BECs with both two- and three-body interactions.