Exact determination of the Peierls-Nabarro frequency

We use a recently developed projection-operator approach to derive the equation of motion for the center of mass of a discrete sine-Gordon (SG) kink, where the center of mass of the kink is represented by the collective variable X. We calculate the small-oscillation frequency of the discrete SG kink trapped inside the Peierls-Nabarro (PN) potential well using an ansatz that introduces the collective variable X into the system and which incorporates discreteness into the kink’s ‘‘shape mode.’’ We obtain essentially exact agreement between molecular-dynamics simulation and theory. We show that when the small-oscillation PN frequency is calculated using the bare ground state of the SG lattice, such as in Ishimori and Munakata’s use of the Keener and McLaughlin perturbation theory, or in our own bare treatment, that the square of the PN frequency is approximately a factor of 2 smaller than values obtained from simulation, even when the kink size is such that discreteness effects are small. In particular, we show that the ratio of the curvature at the bottom of the dressed PN well to the curvature at the bottom of the bare PN well is approximately a factor of 2 for $l_0$>3. We also numerically calculate a first-order dressing for the bare kink, and use this first-order dressed ground state to calculate the small-oscillation PN frequency which agrees with simulation to within 5%. We briefly discuss large-amplitude oscillations of the kink in the PN well.