Modulational and oscillatory instabilities of Bose–Einstein condensates with two- and three-body interactions trapped in an optical lattice potential

• A cubic–quintic Gross–Pitaevskii equation with optical lattice (OL) and harmonic potentials is used.
• We find the stability domain and time-dependent criteria for modulational instability.
• Matter waves are generated through the modulational and oscillatory instabilities for four different possible cases.
• Tuning the strength of OL shrinks and grows the bandwidth of unstable wave numbers.
• In condensates with two- and three-body interactions, oscillatory instability can be realized.

We explain how the modulational and oscillatory instabilities can be generated in Bose–Einstein condensates (BECs) with two- and three-body interactions trapped in a periodic optical lattice with driving harmonic potential. We solve a cubic–quintic Gross–Pitaevskii (GP) equation with external trapping potentials by using both analytical and numerical methods. Using the time-dependent variational approach, we derive and analyze the variational equations for the time evolution of the amplitude and phase of modulational perturbation, and effective potential of the system. Through the effective potential, we obtain the modulational instability condition of the BECs with two- and three-body interactions and shown the effects of the optical potential on the dynamics of the system. We perform direct numerical simulations to support our analytical results, and good agreement is observed.