  We investigate the dynamical instability of Bose-Einstein condensates (BECs)
with higher-order interactions immersed in an optical lattice with weak driving
harmonic potential. For this, we compute both analytically and numerically a
modified Gross-Pitaevskii equation with higher-order nonlinearity and external
potentials generated by magnetic and optical fields. Using the time-dependent
variational approach, we derive the ordinary differential equations for the
time evolution of the amplitude and phase of modulational perturbation. Through
an effective potential, we obtain the modulational instability condition of
BECs and discuss the effect of the higher-order interaction in the dynamics of
the condensates in presence of optical potential. We perform direct numerical
simulations to support our analytical results, and good agreement is found.
