Spontaneous emission of radiation from a discrete sine-Gordon kink

We use a collective-variable approach to study the emission of phonon radiation from a discrete sine-Gordon kink. We find that a kink, trapped and oscillating in the nonlinear Peierls-Nabarro potential well not only radiates phonons smoothly but emits large and sudden bursts of phonon radiation when the frequency of oscillation reaches certain critical values. We show that the bursts occur whenever a harmonic of the kink's oscillation frequency crosses the lower phonon band edge and thus begins to resonate with phonon modes in the high-density-of-states region of the phonon spectrum. We show that the mechanism that explains the radiation bursts and kink motion in the trapped case also accounts for the behavior of a radiating untrapped kink. We introduce into a general discrete kink system the collective variable X, which is the position of the center of the kink, and we derive, using a projection-operator approach we have recently developed, the exact equations of motion, valid for any kink size, for X, and coupled field variables, which dynamically dress the kink and describe the radiation. The equations of motion are valid for any discrete nonlinear Klein-Gordon wave equation that admits a single kink solution. We apply the equations of motion to the discrete sine-Gordon system and obtain excellent agreement with simulation. We also comment on the problems of treating discreteness as a perturbation