Abstract. We study the spatiotemporal dynamics of Bose-Einstein condensates (BECs) with spatiotemporally varying two- and three-body interactions and without an external trapping potential. The cubic and quintic terms in the Gross-Pitaevskii (GP) equation play the roles of external trapping potentials. Spatiotemporal modulation of the two- and three-body interactions can be of great importance in BEC field. But it is the subject of relatively fewer studies. Through theoretical analyses we obtain the equilibrium points (centers and saddles in phase space) of the unperturbed repulsive and attractive cases. With the heteroclinic solution of the unperturbed repulsive system, we theoretically construct the general solution of the 1st-order perturbed equation of the system. By using the boundedness conditions of the general solution we obtain the Melnikov chaos criterion predicting the existence of Smale-horseshoe chaos in the system. With a set of system parameters satisfied the Melnikov chaos criterion, numerical simulations show that the system is in a chaotic state. For BECs that do not satisfy the Melnikov chaos criterion, numerical simulations demonstrate that the modulating frequency can be an effective controlling parameter for the spatiotemporal dynamics of the BECs.

Key words: Bose-Einstein condensates, Spatiotemporal dynamics, Chaos.