We consider the modulational instability (MI) of Bose-Einstein condensate (BEC) described by a modified Gross-Pitaevskii (GP) equation with higher-order nonlinearity both analytically and numerically. A new explicit time-dependent criterion for exciting the MI is obtained. It is shown that the higher-order term can either suppress or enhance the MI, which is interesting for control of the system instability. Importantly, we predict that with the help of the higher-order nonlinearity, the MI can also take place in a BEC with repulsively contact interactions. The analytical results are confirmed by direct numerical simulations.

We study the dynamics of dipolar gas in deep lattices described by a nonlocal nonlinear discrete Gross-Pitaevskii equation. The stabilities and the propagation properties of traveling plane waves in the system with defects are discussed in detail. For a clean lattice, both energetic and dynamical stabilities of the traveling plane waves are studied. It is shown that the system with attractive local interaction can preserve the stabilities, i.e., the dipoles can stabilize the gas because of repulsive nonlocal dipole-dipole interactions. For a lattice with defects, within a two-mode approximation, the propagation properties of traveling plane waves in the system map onto a nonrigid pendulum Hamiltonian with quasimomentum-dependent nonlinearity (induced by the nonlocal interactions). Competition between defects, quasimomentum of the gas, and nonlocal interactions determines the propagation properties of the traveling plane waves. Critical conditions for crossing from a superfluid regime with propagation preserved to a normal regime with defect-induced damping are obtained analytically and confirmed numerically. In particular, the critical conditions for supporting the superfluidity strongly depend on the defect type and the quasimomentum of the plane waves. The nonlocal interaction can significantly enhance the superfluidity of the system.

We present an analytical study on the dynamics of dark solitons in superfluid Fermi gases. By using the modified lens-type transformation, the dynamical equation of superfluid Fermi gases is reduced to a modified onedimensional nonlinear Shorödinger equation (NLSE). Once again, by using the reductive perturbation method, the NLSE is reduced to a standard Korteweg-de Vries equation which may be useful for understanding the dynamics of dark solitons in superfluid Fermi gases. The existence of dark soliton solutions in the Fermi gases is provided. In particular, we show that, by manipulating and controlling the scattering length between Fermi atomics of different components and the external potential, the soliton's parameters (amplitude and width) can be changed in a controllable way.