Open AccessSolitons of the sine-gordon equation coming in clusters
Author(s) -
Cornelia Schiebold
Publication year - 2002
Publication title -
revista matemática complutense
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.093
H-Index - 24
eISSN - 1696-8220
pISSN - 1139-1138
DOI - 10.5209/rev_rema.2002.v15.n1.16969
Subject(s) - mathematics , korteweg–de vries equation , context (archaeology) , sine gordon equation , class (philosophy) , cluster (spacecraft) , mathematical physics , phase (matter) , sine wave , mathematical analysis , soliton , nonlinear system , quantum mechanics , physics , paleontology , artificial intelligence , voltage , computer science , biology , programming language
In the present paper, we construct a particular class of solu- tions 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 _nite 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 e_ect of a phase- shift. The main contribution of this paper is the proof that all this { including an explicit calculation of the phase-shift { an be ex- pressed by concrete asymptotic formulas, which generalize very naturally the known expressions for solitons. Our results con_rm expectations formulated in the context of the Korteweg-de Vries equation by Matveev [17] and Rasinariu et al. [21].In the present paper, we construct a particular class of solu- tions 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 _nite 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 e_ect of a phase- shift. The main contribution of this paper is the proof that all this { including an explicit calculation of the phase-shift { an be ex- pressed by concrete asymptotic formulas, which generalize very naturally the known expressions for solitons. Our results con_rm expectations formulated in the context of the Korteweg-de Vries equation by Matveev [17] and Rasinariu et al. [21]