  The existence of breather type solutions, i.e., periodic in time,
exponentially localized in space solutions, is a very unusual feature for
continuum, nonlinear wave type equations. Following an earlier work [Comm.
Math. Phys. {\bf 302}, 815-841 (2011)], establishing a theorem for the
existence of such structures, we bring to bear a combination of
analysis-inspired numerical tools that permit the construction of such wave
forms to a desired numerical accuracy. In addition, this enables us to explore
their numerical stability. Our computations show that for the spatially
heterogeneous form of the $\phi^4$ model considered herein, the breather
solutions are generically found to be unstable. Their instability seems to
generically favor the motion of the relevant structures. We expect that these
results may inspire further studies towards the identification of stable
continuous breathers in spatially-heterogeneous, continuum nonlinear wave
equation models.
