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

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