Pulse evolution for a two-dimensional Sine-Gordon equation

The evolution lump and ring solutions of a Sine-Gordon equation in two-spac e dimensions is considered. Approximate equations governing this evolution are derived using a pulse or ring with variable parameters in an averaged L agrangian 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. How ever, 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 considere d. It is shown both asymptotically and numerically that the angular depende nce produces spiral waves which shed angular momentum, leading to the ultim ate collapse of the pulse. Good quantitative agreement between the asymptot ic and numerical solutions is found. In addition, it is shown how the resul ts of the present work can be applied to the Baby Skyrme model. In this reg ard, it is shown how the non-zero degree of solutions of the Baby Skyrme mo del prevents the collapse of a non-zero degree pulse shedding zero degree r adiation. 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 c omplicated behaviour due to radiation of angular momentum can be captured i n simple terms by approximate equations for the relevant degrees of freedom . (C) 2001 Elsevier Science B.V. All rights reserved.