  The stability of nonstationary states of homogeneous spin-1 Bose-Einstein
condensates is studied by performing Bogoliubov analysis in a frame of
reference where the state is stationary. In particular, the effect of an
external magnetic field is examined. It is found that a nonzero magnetic field
introduces instability in a $^{23}$Na condensate. The wavelengths of this
instability can be controlled by tuning the strength of the magnetic field. In
a $^{87}$Rb condensate this instability is present already at zero magnetic
field. Furthermore, an analytical bound for the size of a stable condensate is
found, and a condition for the validity of the single-mode approximation is
presented. Realization of the system in a toroidal trap is discussed and the
full time development is simulated.
