Full paper in PDF:

$%C. Schiebold, Solitons of the Sine-Gordon equation coming in clusters, Rev. Mat. Complut. 15
(2002), no. 1, 265–325.%$

Solitons of the Sine-Gordon Equation
Coming in Clusters

C. SCHIEBOLD

Fakultät für Mathematik und Informatik

Universität Jena

Ernst-Abbe-Platz 1-4

D-07743 Jena — Germany

Universität Jena

Ernst-Abbe-Platz 1-4

D-07743 Jena — Germany

Revised: May 3, 2001

ABSTRACT

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.