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

TL;DR

This paper introduces a bi-Hamiltonian framework for multipeakon solutions of the Camassa-Holm equation and proposes a novel method for their dissipative prolongations post-collision, preserving momentum.

Contribution

It provides an explicit bi-Hamiltonian formulation and a new approach to extend multipeakons after collisions while maintaining key invariants.

Findings

Multipeakons can be extended after collision with reduced peakons.

The momentum of peakons is preserved during dissipative prolongation.

A novel method for prolonging solutions post-collision is proposed.

Abstract

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.