Solitons of the sine-Gordon equation coming in clusters

• Autores: C. Schiebold
• Localización: Revista matemática complutense, ISSN-e 1988-2807, ISSN 1139-1138, Vol. 15, Nº 1, 2002, págs. 265-325
• Idioma: inglés
• DOI: 10.5209/rev_rema.2002.v15.n1.16969
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• Resumen

  • 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].