Pulse evolution for a two-dimensional Sine-Gordon equation

The evolution lump and ring solutions of a Sine-Gordon equation in two-space dimensions is considered. Approximate equations governing this evolution are derived using a pulse or ring with variable parameters in an averaged Lagrangian for the Sine-Gordon equation. It was found by Neu [Physica D 43 (1990) 421] that angular variations of the pulse shape may stabilise it. However, no study of the radiation produced by the pulse was available. In the present work, the coupling of the pulse to the shed radiation is considered. It is shown both asymptotically and numerically that the angular dependence produces spiral waves which shed angular momentum, leading to the ultimate collapse of the pulse. Good quantitative agreement between the asymptotic and numerical solutions is found. In addition, it is shown how the results of the present work can be applied to the Baby Skyrme model. In this regard, it is shown how the non-zero degree of solutions of the Baby Skyrme model prevents the collapse of a non-zero degree pulse shedding zero degree radiation. It is also indicated how the present results could be applied to the study of vortex models. The analysis presented in this work shows how complicated behaviour due to radiation of angular momentum can be captured in simple terms by approximate equations for the relevant degrees of freedom.