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

Multipeakons are special solutions to the Camassa-Holm equation 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.