Solitons of the sine-gordon equation coming in clusters

Abstract

In the present paper, we construct a particular class of solu- tions of the sine-Gordon equation, which is the exact analogue of the so-called negatons, a solution class of the Korteweg-de Vries equation discussed by Matveev [17] and Rasinariu et al. [21]. Their characteristic properties are: Each solution consists of a _nite number of clusters. Roughly speaking, in such a cluster solitons are grouped around a center, and the distance between two of them grows logarithmically. The clusters themselves rather behave like solitons. Moving with constant velocity, they collide elastically with the only e_ect of a phase- shift. The main contribution of this paper is the proof that all this { including an explicit calculation of the phase-shift { an be ex- pressed by concrete asymptotic formulas, which generalize very naturally the known expressions for solitons. Our results con_rm expectations formulated in the context of the Korteweg-de Vries equation by Matveev [17] and Rasinariu et al. [21].