We analyzed the stability for a BEC within the framework of the discrete model(6)and required that the left and right incident superfluid currents have to be stable. Their stability is determined by the stability of the corresponding Bogoliubov phonons on an infinite homogeneous lattice (that is, without applied removal of atoms). The stability of the homogeneous lattice is found using the substitution(7)where ρ > 0 characterizes the uniform density, and |v_n| , |w_n| ≪ ρ are small perturbations. Linearizing Eq. 6 (with γ = 0) with respect to v_n and w_n, we found two dispersion branches(8)Consider now a positive scattering length, U > 0, which corresponds to the experiments reported here. One can then identify the stability domain for Bogoliubov phonons and, hence, the stability of the superfluid current, requiring ω_± to be real for the given q and all real k. This results in the constraint 0 ≤ q < π/2, that is, only slow currents are dynamically stable.