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$%C. Schiebold, Solitons of the Sine-Gordon equation coming in clusters, Rev. Mat. Complut. 15 (2002), no. 1, 265–325.%$
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 (1994) and Rasinariu et al. (1996) 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 (1994) and Rasinariu et al. (1996).
2000 Mathematics Subject Classification: 15A90, 35Q51, 35Q53, 37K40.