Lattice dynamics of solitary wave excitations

In this work we study the dynamical behavior of solitonic excitations in lattice systems. This behavior entails dramatic differences from the continuum-like picture of a merrily propagating soliton. In fact, the solitary wave motion is damped (for appropriate conditions even exponentially) and results in the eventual trapping and pinning of the soliton between two sites of the lattice. We trace this behavior to the nonlinear resonant damping of the coherent structure eigenmodes and their harmonics (generated by nonlinearity) with the extended wave modes that can be sustained by the lattice. To clarify this picture we develop the discrete Evans function and use it in conjunction with linear stability analysis to numerically investigate the soliton stability. The findings are verified analytically by means of singular perturbation theory, the continuum Evans function technique and asymptotics beyond all orders. Once the stability picture is established, Hamiltonian dispersive normal form theory is used to identify the resonant damping of the soliton internal modes with the modes of the essential spectrum (the extended waves). In this way, the decay rates of the soliton energy as well as the asymptotic state of the pattern are obtained. All the results are presented in the context of the discrete sine Gordon and &phis;4 models (with occasional excursions demonstrating the relevant analogies in the case of the nonlinear Schrodinger equation). The behavior is analyzed as a function of the lattice spacing (essentially the inverse of what we will call the discreteness parameter) and the smooth limiting process to the continuum case is demonstrated. All results are in very good agreement with our numerical experiments.