Wobbles and other kink-breather solutions of the sine-Gordon model

We study various solutions of the sine-Gordon model in $\left(1+1\right)$ dimensions. We use the Hirota method to construct some of them and then show that the wobble, discussed in detail in a recent paper by Kälberman, is one of such solutions. We concentrate our attention on a kink and its bound states with one or two breathers. We study their stability and some aspects of their scattering properties on potential wells and on fixed boundary conditions.

Figure 1

Energy as a function of time as seen in a simulation started with $\lambda =1.15$

Figure 2

Field configurations seen in a simulation involving a wobble scattering on a potential hole (of depth 0.1 and located in $-15<x<15$). $t=\left(\text{a}\right)60$ and (b) 450.

Figure 3

Energies as a function of time as seen when the starting file was given by (12): (a) total and (b) potential energy.

Figure 4

Field configurations at $t=\left(\text{a}\right)6750$ and (b) 6753.