Multipeakons viewed as geodesics

Tomasz Cieślak¹ ; Michał Gaczkowski¹ ; Mariusz Kubkowski² ; Marcin Małogrosz¹

¹ Institute of Mathematics Polish Academy of Sciences 00-656 Warszawa, Poland
² Wydział Matematyki i Nauk Informacyjnych Politechnika Warszawska 00-662 Warszawa, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 65 (2017) no. 2, pp. 153-164.

Résumé

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.

Détail

DOI : 10.4064/ba8119-6-2017

Keywords: adress problem qualitative properties multipeakons particular solutions camassa holm equation approach makes well known evolution multipeakons governed geodesic motion particle n dimensional surface whose metric tensor given via inverse matrix defining hamiltonian approach yields properties twopeakons simple classify initial shapes twopeakons according occurrence collision moreover extend class matrices invertible similar reasons occurring hamiltonian get exact formulas inverses