0 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 −1 , where h > 0 is the lattice spacing. The specific cases we consider in detail are the discrete sine-Gordon (SG) and discrete φ 4 models. For finite d and in the continuum limit (d → ∞) 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 kink, initially propagating in the lattice dynamically adjusts in the absence of an available family of traveling kinks. We study in detail the final stages of the discrete kink's evolution during which it is pinned to a specified lattice site (equilibrium position in the Peierls-Nabarro barrier). We find: (i) for d sufficiently large (sufficiently small lattice spacing), the state of the system approaches an asymptotically stable ground state static kink (centered between lattice sites). (ii) for d sufficiently small d < d * the static kink bifurcates to one or more time periodic states. For the discrete φ 4 we have: wobbling kinks which have the same spatial symmetry as the static kink as well as " g-wobblers " and " e-wobblers " , which have different spatial symmetry. In the discrete sine-Gordon case, the " e-wobbler " has the spatial symmetry of the kink whereas the " g-wobbler " has the opposite one. These time-periodic states may be regarded as a class of discrete breather / topological defect states; they are spatially localized and time periodic oscillations mounted on a static kink background. The large time limit of solutions with initial data near a kink is marked by damped oscillation about one of these two types of asymptotic states. In case (i) we compute the characteristics of the damped oscillation (frequency and d-dependent rate of decay). In case (ii) we prove the existence of, and give analytical and numerical evidence for the asymptotic stability of wobbling solutions. 1 The mechanism for decay is the radiation of excess energy, stored in internal modes, away from the kink core to infinity. This process is studied in detail using general techniques of scattering theory and normal forms. In particular, we derive a dispersive …