On kink-dynamics of the perturbed sine-Gordon equation

The dynamics of 2πn-kink solutions to the perturbed sine-Gordon equation (PSGE), propagating with velocity c near unity is investigated. Using qualitative methods of differential equation theory and based on numerical simulations, we find that the dependence of the propagation velocity c on the bias parameter γ has a spiral-like form in the (c, γ)-plane in the neighborhood c = 1 for all types of 2πn-kink solutions for appropriate values of the loss parameters in the PSGE. We find numerically that the γ-coordinates of the focal points, A ⁱ of these "spirals" have a scaling property. So, it is possible to estimate the lower boundary of the parameter region where the 2πn-kink solutions to the PSGE can exist. The phase space structure at the points A ⁱ for the corresponding ODE system is also investigated. The form of 2πm-kink solutions in the neighborhood of the points A i is explained and the dynamics is discussed. A certain combination of the dissipative parameters of the PSGE is shown to be essential. The dependence of the height of the zero field step of the long Josephson junction modeled by the PSGE is also obtained.