Discreteness effects on the double-quadratic kink

We study the static and dynamic properties of a kink in a chain of harmonically coupled atoms on a double-quadratic substrate. We treat intrinsically the lattice discreteness without approximation and demonstrate that the stable kink does not cause a phase shift of the phonons, and relate this result to Levinson's theorem. Using a recently developed projection-operator approach, we derive exact equations of motion for the kink center of mass, X, and coupled field variables. With neglect of radiation, a zeroth-order expression is obtained for the frequency with which the trapped kink oscillates in the Peierls-Nabarro well, and we show that the frequency lies in the phonon band. Consequently, we show that the effects of discreteness on the double-quadratic kink manifest themselves in surprisingly different ways than in a typical discrete kink-bearing system, i.e., the center-of-mass motion of a trapped double-quadratic kink is a quasimode in the same sense as is the shape mode of the sine-Gordon kink [R. Boesch and C. R. Willis, Phys. Rev. B 42, 2290 (1990)]. We solve numerically the collective-variable equations of motion for the trapped and untrapped regimes of the discrete kink motion, and compare the results to those found for various other models.