Interaction of sine‐Gordon solitons in the model with attracting impurities

We study properties of the localized solutions to the sine-Gordon equation excited on the attractive impurity by a moving kink. The cases of one-dimensional and two-dimensional spatially extended impurities are considered. For the case of one-dimensional impurity the possibility of excitation of the first even and odd high-amplitude impurity modes by the moving kink is demonstrated. By linearizing the sine-Gordon equation the dispersion relations for the small-amplitude localized impurity modes were obtained. The numerically obtained dispersion relations in the case of low oscillation amplitudes are in a good agreement with the results of analytical calculations. For the case of two-dimensional impurity we show the possibility of excitation of the nonlinear high-amplitude waves of new type called here a breathing pulson and a breathing 2D soliton. We suggest analytical expressions to model these nonlinear excitations. The breathing pulson and breathing 2D soliton are long-lived and can be of both symmetric and asymmetric type depending on the impurity type. The range of the impurity parameters where the breathing pulson and breathing 2D soliton can be excited was determined. The dependencies of the oscillation frequency and the amplitude of the excited impurity modes on the impurity parameters are reported. Copyright (C) 2016 John Wiley & Sons, Ltd.