Rotationally symmetric numerical solutions to the sine-Gordon equation

We examine numerically the properties of solutions to the spherically symmetric sine-Gordon equation given an initial profile which coincides with the one-dimensional breather solution and refer to such solutions as ring waves. Expanding ring waves either exhibit a return effect or expand towards infinity. This can be explained by means of a perturbation approach. For a moderate initial radius of the shrinking ring wave we find an evolution of pulson modes. The ring waves are shown to survive the interaction between other ring waves.