  We explore the dynamical stabilities of a quasi-one dimensional (1D)
Bose-Einstein condensate (BEC) consisting of fixed $N$ atoms with
time-independent external potential. For the stationary states with zero flow
density the general solution of the perturbed time evolution equation is
constructed, and the stability criterions concerning the initial conditions and
system parameters are established. Taking the lattice potential case as an
example, the stability and instability regions on the parameter space are
found. The results suggest a method for selecting experimental parameters and
adjusting initial conditions to suppress the instabilities.
