  We consider the instability and stability of periodic stationary solutions to
the classical \phi^4 equation numerically. In the superluminal regime, the
model possesses dnoidal and cnoidal waves. The former are modulationally
unstable and the spectrum forms a figure eight intersecting at the origin of
the spectral plane. The latter can be modulationally stable and the spectrum
near the origin in that case is represented by vertical bands along the purely
imaginary axis. The instability of the cnoidal states in that case stems from
elliptical bands of complex eigenvalues far from the spectral plane origin. In
the subluminal regime, there exist only snoidal waves which are modulationally
unstable. Considering the subharmonic perturbations, we show that the snoidal
waves in the subluminal regime are spectrally unstable with respect to all
subharmonic perturbations, while for the dnoidal and cnoidal waves in the
superluminal regime, the transition between the spectrally stable state and the
spectrally unstable state occurs through a Hamiltonian Hopf bifurcation. The
dynamical evolution of the unstable states is also considered, leading to some
interesting spatio-temporal localization events.
