Modulation Theory for Radially Symmetric Kink Waves Governed by a Multi-Dimensional Sine-Gordon Equation

We derive a modulation theory for the resolution of radially symmetric kink waves governed by a multi-dimensional sine-Gordon equation. Whitham modulation theory is developed to explain the return of an expanding kink wave, as well as predicting its maximum expansion radius and its return time. Comparisons with full numerical solutions of the sine-Gordon equation show that the modulation theory gives excellent predictions for not only the returning time and the maximum expansion radius, but also for the details of the kink itself. In addition, the method can be extended to dissipative sine-Gordon equations and generalized to deal with a wide class of initial conditions beyond kinks.