Multipeakons viewed as geodesics

Volume 65 / 2017

Tomasz Cieślak, Michał Gaczkowski, Mariusz Kubkowski, Marcin Małogrosz Bulletin Polish Acad. Sci. Math. 65 (2017), 153-164 MSC: 53A17, 15A09. DOI: 10.4064/ba8119-6-2017 Published online: 19 July 2017

Abstract

We adress the problem of qualitative properties of multipeakons, particular solutions of the Camassa–Holm equation. Our approach makes use of the well-known fact that the evolution of multipeakons is governed by the geodesic motion of a particle on an $N$-dimensional surface whose metric tensor is given via the inverse matrix to the one defining the Hamiltonian. Our approach yields some properties of twopeakons in a very simple way. We classify initial shapes of twopeakons according to the occurrence of collision. Moreover we extend the class of matrices that are invertible for similar reasons to the one occurring in the Hamiltonian. We get exact formulas for the inverses.

Authors

  • Tomasz CieślakInstitute of Mathematics
    Polish Academy of Sciences
    00-656 Warszawa, Poland
    e-mail
  • Michał GaczkowskiInstitute of Mathematics
    Polish Academy of Sciences
    00-656 Warszawa, Poland
    e-mail
  • Mariusz KubkowskiWydział Matematyki i Nauk Informacyjnych
    Politechnika Warszawska
    00-662 Warszawa, Poland
    e-mail
  • Marcin MałogroszInstitute of Mathematics
    Polish Academy of Sciences
    00-656 Warszawa, Poland
    e-mail

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