Discreteness effects on the double sine-Gordon kink

We derive and analyze the exact collective variable equations of motion for the discrete double sine-Gordon (DSG) equation as a typical example of a discrete Klein-Gordon equation with an internal mode. In the continuum DSG system, the small-amplitude equations of motion for the center of mass X(t) and internal mode R(t) are uncoupled, but discreteness causes the center of mass mode X(t) and the internal mode R(t) to be strongly coupled, leading to further qualitative phenomena in addition to the known discreteness phenomena such as the Peierls-Nabarro potential and effects due to internal modes. We show that the behavior of large kinks where the distance between subkinks is large compared to the size of the subkink is different from the behavior of small kinks where the distance between subkinks is smaller than the size of the subkink. In the case of the small kink, we show that there are two minima per unit cell instead of the usual one minimum and that the discrete DSG kink becomes temporarily trapped and untrapped due to conversion of center of mass energy to internal mode energy and back until the kink finally becomes permanently trapped. We show by simulations that the radiation occurs in bursts and decreases sharply at specific frequencies, which can be straightforwardly explained by collective variable theory. In addition, the asymmetry of the phonon radiation by the moving DSG kink is responsible for asymmetries in the subkink-subkink interaction, causing phenomena such as excitation of the internal mode and causing the subkinks to travel sometimes at different speeds. We determine the exact Peierls-Nabarro and internal mode small oscillation frequencies.