Dynamical properties of a kink of the Sine-Gordon equation trapped in a potential well

The dynamical properties of a kink of the Sine-Gordon equation trapped in a potential well and interacting with a periodic spatial inhomogeneity are investigated. It is shown that the description of the kink motion obtained by the adiabatic approximation breaks down. This fact is explained in term of changes in the kink form due to the presence of such perturbations. We will show that in the presence of spatial periodic inhomogeneity there are parameter ranges where complex behaviour of the kink dynamics is observed. Moreover, when the spatial periodic perturbation is switched off for each kink initial velocity the radiation emission corresponding to well defined wave number is inhibited.