Multipeakons viewed as geodesics
Volume 65 / 2017
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.
