# Wydawnictwa / Czasopisma IMPAN / Bulletin Polish Acad. Sci. Math. / Wszystkie zeszyty

## Multipeakons viewed as geodesics

### Tom 65 / 2017

Bulletin Polish Acad. Sci. Math. 65 (2017), 153-164 MSC: 53A17, 15A09. DOI: 10.4064/ba8119-6-2017 Opublikowany online: 19 July 2017

#### Streszczenie

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.

#### Autorzy

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