in Titre de série : | Physica D, Vol.18 No.3 | Titre : | The solitons of Zabusky and Kruskal revisited: Perspective in terms of the periodic spectral transform | Type de document : | articles et extraits | Auteurs : | A. R. Osborne, Auteur ; L. Bergamasco, Auteur | Année de publication : | 1986 | Importance : | p.26-46 | ISBN/ISSN/EAN : | 0167-2789 | Langues : | Français (fre) | Résumé : | We consider the numerical soliton experiments of Zabusky and Kruskal [1] and address their results in light of the spectral structure of the periodic Korteweg-deVries (KdV) equation. We derive and exploit a simple relationship between the infinite-line transfer matrix and the monodromy matrix of periodic theory; this establishes the connection, at all stages of our analysis, between the infinite-line soliton and the soliton limit of periodic theory. We develop analytical and numerical procedures for determining the spectral structure of a periodic wave train and discuss how the methods may be viewed as a generalization of ordinary periodic Fourier analysis. Finally we introduce the concept of a ""reference level"" which is defined to be the level upon which the solitons propagate. These results, together with judicious numerical experiments, allow us to reassess the work of Zabusky and Kruskal from the standpoint of periodic theory for the KdV equation. We find their system to be soliton dominated and our estimates of the soliton amplitudes and phase speeds agree well with those given in their classic paper. We also find that, even in the presence of periodic boundary conditions, the solitons behave like infinite-line solitons in that they undergo phase shifts upon collision with each other. Our estimate of the FPU recurrence time, which includes the effect of phase shifting, is within one percent of their observed value.
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in Physica D, Vol.18 No.3. The solitons of Zabusky and Kruskal revisited: Perspective in terms of the periodic spectral transform [articles et extraits] / A. R. Osborne, Auteur ; L. Bergamasco, Auteur . - 1986 . - p.26-46. ISSN : 0167-2789 Langues : Français ( fre) Résumé : | We consider the numerical soliton experiments of Zabusky and Kruskal [1] and address their results in light of the spectral structure of the periodic Korteweg-deVries (KdV) equation. We derive and exploit a simple relationship between the infinite-line transfer matrix and the monodromy matrix of periodic theory; this establishes the connection, at all stages of our analysis, between the infinite-line soliton and the soliton limit of periodic theory. We develop analytical and numerical procedures for determining the spectral structure of a periodic wave train and discuss how the methods may be viewed as a generalization of ordinary periodic Fourier analysis. Finally we introduce the concept of a ""reference level"" which is defined to be the level upon which the solitons propagate. These results, together with judicious numerical experiments, allow us to reassess the work of Zabusky and Kruskal from the standpoint of periodic theory for the KdV equation. We find their system to be soliton dominated and our estimates of the soliton amplitudes and phase speeds agree well with those given in their classic paper. We also find that, even in the presence of periodic boundary conditions, the solitons behave like infinite-line solitons in that they undergo phase shifts upon collision with each other. Our estimate of the FPU recurrence time, which includes the effect of phase shifting, is within one percent of their observed value.
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