A variational approach to the modulational-oscillatory instability of Bose-Einstein condensates in an optical potential

We use the time-dependent variational approach to demonstrate how the modulational and oscillatory instabilities can be generated in Bose-Einstein condensates (BECs) trapped in a periodic optical lattice with weak driving harmonic potential. We derive and analyze the ordinary differential equations for the time evolution of the amplitude and phase of the modulational perturbation, and obtain the instability condition of the condensates through the effective potential. The effect of the optical potential on the dynamics of the BECs is shown. We perform direct numerical simulations to support our theoretical findings, and good agreement is found.