# Solitons of the sine-Gordon equation coming in clusters.

• Volume: 15, Issue: 1, page 265-325
• ISSN: 1139-1138

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## Abstract

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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 solutions.Our results confirm expectations formulated in the context of the Korteweg-de Vries equation by Matveev (1994) and Rasinariu et al. (1996).

## How to cite

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Schiebold, Cornelia. "Solitons of the sine-Gordon equation coming in clusters.." Revista Matemática Complutense 15.1 (2002): 265-325. <http://eudml.org/doc/44401>.

@article{Schiebold2002,
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 [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 solutions.Our results confirm expectations formulated in the context of the Korteweg-de Vries equation by Matveev (1994) and Rasinariu et al. (1996).},
author = {Schiebold, Cornelia},
journal = {Revista Matemática Complutense},
keywords = {Ecuaciones diferenciales en derivadas parciales; Solitones; Ecuaciones de evolución no lineales; exact solutions; negatons; cluster; phase-shift; asymptotic formulas},
language = {eng},
number = {1},
pages = {265-325},
title = {Solitons of the sine-Gordon equation coming in clusters.},
url = {http://eudml.org/doc/44401},
volume = {15},
year = {2002},
}

TY - JOUR
AU - Schiebold, Cornelia
TI - Solitons of the sine-Gordon equation coming in clusters.
JO - Revista Matemática Complutense
PY - 2002
VL - 15
IS - 1
SP - 265
EP - 325
AB - 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 solutions.Our results confirm expectations formulated in the context of the Korteweg-de Vries equation by Matveev (1994) and Rasinariu et al. (1996).
LA - eng
KW - Ecuaciones diferenciales en derivadas parciales; Solitones; Ecuaciones de evolución no lineales; exact solutions; negatons; cluster; phase-shift; asymptotic formulas
UR - http://eudml.org/doc/44401
ER -

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