Multi-component generalizations of the CH equation:Geometrical aspects, peakons and numerical examples

The Lax pair formulation of the two-component Camassa-Holm equation (CH2) is generalized to produce an integrable multi-component family, CH(n, k), of equations with n components and 1 ≤ |k| ≤ n velocities. All of the members of the CH(n, k) family show fluid-dynamics properties with coherent solitons following particle characteristics. We determine their Lie- Poisson Hamiltonian structures and give numerical examples of their soliton solution behaviour. We concentrate on the CH(2, k) family with one or two velocities, including the CH(2, -1) equation in the Dym position of the CH2 hierarchy. A brief discussion of the CH(3, 1) system reveals the underlying graded Lie-algebraic structure of the Hamiltonian formulation for CH(n, k) when n ≥ 3.