Dissipative prolongations of the multipeakon solutions to the Camassa–Holm equation

Abstract

Multipeakons are special solutions to the Camassa–Holm equation. They are described by an integrable geodesic flow on a Riemannian manifold. We present a bi-Hamiltonian formulation of the system explicitly and write down formulae for the associated first integrals. Then we exploit the first integrals and present a novel approach to the problem of the dissipative prolongations of multipeakons after the collision time. We prove that an n-peakon after a collision becomes an n−1-peakon for which the momentum is preserved.