  We have found exact, periodic, time-dependent solitary wave solutions of a
discrete $\phi^4$ field theory model. For finite lattices, depending on whether
one is considering a repulsive or attractive case, the solutions are either
Jacobi elliptic functions $\sn(x,m)$ (which reduce to the kink function
$\tanh(x)$ for $m\to 1$), or they are $\dn(x,m)$ and $\cn(x,m)$ (which reduce
to the pulse function $\sech(x)$ for $m\to 1$). We have studied the stability
of these solutions numerically, and we find that our solutions are linearly
stable in most cases. We show that this model is a Hamiltonian system, and that
the effective Peierls-Nabarro barrier due to discreteness is zero not only for
the two localized modes but even for all three periodic solutions. We also
present results of numerical simulations of scattering of kink--anti-kink and
pulse--anti-pulse solitary wave solutions.
