Dynamics of lattice kinks

We consider a class of Hamiltonian nonlinear wave equations governing a fie ld defined on a spatially discrete one-dimensional lattice, with discretene ss parameter, d = h(-1), where h > 0 is the lattice spacing. The specific c ases we consider in detail are the discrete sine-Gordon (SG) and discrete p hi(4) models. For finite d and in the continuum limit (d --> infinity) thes e equations have static kink-like (heteroclinic) states which are stable. I n contrast to the continuum case, due to the breaking of Lorentz invariance , discrete kinks cannot be "Lorentz boosted" to obtain traveling discrete k inks. Peyrard and Kruskal pioneered the study of how a kink, initially prop agating in the lattice, dynamically adjusts in the absence of an available family of traveling kinks. We study in detail the final stages of the discr ete kink's evolution during which it is pinned to a specified lattice site (equilibrium position in the Peierls-Nabarro barrier). We find the followin g: 1. 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). 2. For d sufficiently small, d < d(*), the static kink bifurcates to one or more time-periodic states. For the discrete phi(4) we have wobbling kinks which have the same spatial symmetry as the static kink as well as "g-wobbl ers" and "e-wobblers", which have different spatial symmetry. In the discre te SG case, the "e-wobbler" has the spatial symmetry of the kink, whereas t he "g-wobbler" has the opposite one. These time-periodic states may be rega rded as a class of discrete breather/topological defect states; they are sp atially localized and time-periodic oscillations mounted on a static kink b ackground. The large time limit of solutions with initial data near a kink is marked b y damped oscillation about one of these two types of asymptotic states. In case (1) we compute the characteristics of the damped oscillation (frequenc y and d-dependent rate of decay). In case (2) we prove the existence of, an d give analytical and numerical evidence for the asymptotic stability of wo bbling solutions. The mechanism for decay is the radiation of excess energy , stored in internal modes, away from the kink core to infinity. This proce ss is studied in detail using general techniques of scattering theory and n ormal forms. In particular, we derive a dispersive normal form, from which one can anticipate the character of the dynamics. The methods we use are ve ry general and are appropriate for the study of dynamical systems which may be viewed as a system of dis crete oscillators (e.g. kink together with it s internal modes) coupled to a field (e.g. dispersive radiation or phonons) . The approach is based on and extends an approach of one of the authors (M IW) and Soffer in previous work. Changes in the character of the dynamics, as d varies, are manifested in topological changes in the phase portrait of the normal form. These changes are due to changes in the types of resonanc es which occur among the discrete internal modes and the continuum radiatio n modes as Li varies. Though derived from a time-reversible dynamical syste m, this normal form has a dissipative character. The dissipation is of an i nternal nature, and corresponds to the transfer of energy from the discrete to continuum radiation modes. The coefficients which characterize the time -scale of damping (or lifetime of the internal mode oscillations) are a non linear analogs of "Fermi's golden rule", which arises in the theory of spon taneous emission in quantum physics. (C) 2000 Elsevier Science B.V. All rig hts reserved.