Sine-Gordon kinks on a discrete lattice. II. Static properties

The differences in the static properties between the discrete and continuum kink solutions of the one-dimensional sine-Gordon equation are found with use of a discretized Hamiltonian formalism which we have developed previously. The difference in the kink profile is treated as a static dressing of the continuum soliton. We point out some of the salient features of the static dressing and their dependence on the kink-size parameter $l_0$. We find that the dressing has important effects on dynamical quantities such as the kink pinning frequency and the depth of the Peierls-Nabarro potential. When the static dressing is included, the actual depth and curvature of the potential are found to be approximately twice the value obtained using the continuum approximation. Hence, the square of the pinning frequency is doubled. We also find that for short kinks (small $l_0$) the static dressing develops some spatial structure which we believe will have important consequences in the dynamics, namely in the spontaneous radiation of phonons by the moving kink. We show that our analytic results agree closely with results obtained from molecular-dynamics simulations. The agreement with the simulations provides strong evidence for the reliability of our formalism to predict manifestations of discreteness and their ramifications.