Resumen de Solitons of the sine-Gordon equation coming in clusters

C. Schiebold

• In the present paper, we construct a particular class of solutions 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 finite 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 effect of a phase-shift.

  The main contribution of this paper is the proof that all this -including an explicit calculation of the phase-shift - can be expressed by concrete asymptotic formulas, which generalize very naturally the known expressions for solitons.

  Our results confirm expectations formulated in the context of the Korteweg-de Vries equation by Matveev [17] and Rasinariu et al. [21].