Landau and dynamical instabilities of Bose-Einstein condensates in a Kronig-Penney potential

We study elementary excitations of Bose-Einstein condensates in a one-dimensional periodic potential and discuss the stability of superfluid flow based on the Kronig-Penney model. We analytically solve the Bogoliubov equations and calculate the excitation spectrum. The Landau and dynamical instabilities occur in the first condensate band when the superfluid velocity exceeds certain critical values as in a sinusoidal potential. It is found that the onset of the Landau instability coincides with the point where the perfect transmission of low-energy excitations is forbidden, while the dynamical instability occurs when the effective mass is negative. The condensate band has a swallow-tail structure when the periodic potential is shallow compared to the mean-field energy. We find that the upper side of a swallow-tail is dynamically unstable although the excitation spectrum has a linear dispersion reflecting the positive effective mass.