2010/09/28 by D. D. Holm, R. I. Ivanov
Physics and Astronomy · Mathematics · #nlin.SI #math-ph #math.AP #math.MP
paper · pdf · doi:10.1088/1751-8113/43/49/492001
published as J. Phys. A: Math. Theor. 43 No 49 (10 December 2010) 492001 (20pp) · 19 pages 5 figures comments are welcome
arxiv created 2010/09/28 · arxiv updated 2015/05/20
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.