Dynamics of Lattice Kinks

We consider a class of Hamiltonian nonlinear wave equations governing a field defined on a spatially discrete one dimensional lattice, with discreteness parameter, d = h sup (-1), where h > 0 is the lattice spacing. The specific cases we consider in detail are the discrete sine-Gordon (SG) and discrete /0 sup 4 models. For finite d and in the continuum limit (d -> inf) these equations have static kink-like (heteroclinic) states which are stable. In contrast to the continuum case, due to the breaking of Lorentz invariance, discrete kinks cannot be "Lorentz boosted" to obtain traveling discrete kinks. Peyrard and Kruskal pioneered the study of how a kin, initially propagating in the lattice dynamically adjusts in the absence of an available family of traveling kinds.