DIMACS TR: 99-51

Authors: P.G. Kevrekidis and M.I. Weinstein

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 $\phi^4$ models. For finite $d$ and in the continuum limit ($d\to\infty$) 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

For discrete SG and discrete $\phi^4$ 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.

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.

The mechanism for decay is the radiation of excess energy, stored in {\it 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 {\it dispersive normal form}, from which one can anticipate the character of the dynamics. The methods we use are very general and are appropriate for the study of dynamical systems which may be viewed as a system of discrete oscillators ({\it e.g.} kink together with its {\it internal modes}) coupled to a field ({\it e.g.} dispersive radiation or phonons). The approach is based on and extends an approach of one of the authors (MIW) and A. 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 resonances which occur among the discrete internal modes and the continuum radiation modes, as $d$ varies.

Though derived from a time-reversible dynamical system, this normal form has a dissipative character. The dissipation is of an internal 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 {\it lifetime} of the internal mode oscillations) are a nonlinear analogue of ``Fermi's golden rule", which arises in the theory of spontaneous emission in quantum physics.

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