Exact solitary wave solutions for a discrete $\lambda \phi^4$ field   theory in 1+1 dimensions
  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. 
Fred Cooper, Avinash Khare, Bogdan Mihaila, and Avadh Saxena