Multi-component generalizations of the {CH} equation: Geometrical Aspects, Peakons and Numerical Examples

Abstract

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 $1\le |k|\le n$ 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 $n\ge 3$.

19 pages 5 figures comments are welcome

References in corpus (11)

• On an integrable two-component Camassa-Holm shallow water system
• Inverse Scattering Transform for the Camassa-Holm equation
• On asymptotically equivalent shallow water wave equations
• Singular solutions of a modified two-component Camassa-Holm equation
• The momentum map representation of images
• Poisson structure and Action-Angle variables for the Camassa-Holm equation
• Geodesic flows on semidirect-product Lie groups: geometry of singular measure-valued solutions
• The geodesic Vlasov equation and its integrable moment closures
• Smooth and Peaked Solitons of the CH equation
• Geometric dynamics of Vlasov kinetic theory and its moments
• Hamiltonian Structures and Reciprocal Transformations for the $r$-KdV-CH Hierarchy

Cited by in corpus (20)

• Two-component {CH} system: Inverse Scattering, Peakons and Geometry
• A synthetical two-component model with peakon solutions
• Negative order KdV equation with both solitons and kink wave solutions
• A new two-component integrable system with peakon solutions
• Novikov algebras and a classification of multicomponent Camassa-Holm equations
• Waltzing peakons and compacton pairs in a cross-coupled Camassa-Holm equation
• Conserved quantities, continuation and compactly supported solutions of some shallow water models
• Multisoliton solutions of the two-component Camassa-Holm system and their reductions
• Vlasov moment flows and geodesics on the Jacobi group
• Applications of Nijenhuis Geometry IV: multicomponent KdV and Camassa-Holm equations
• On a Lagrangian reduction and a deformation of completely integrable systems
• A Lagrangian view on complete integrability of the two-component Camassa-Holm system
• Real-valued algebro-geometric solutions of the two-component Camassa-Holm hierarchy
• Geometrical demonstration for persistence properties for a bi-Hamiltonian shallow water system
• Matrix G-Strands
• Integrable negative flows of the Heisenberg ferromagnet equation hierarchy
• The Two-Component Camassa-Holm Equations CH(2,1) and CH(2,2): First-Order Integrating Factors and Conservation Laws
• A generalised multicomponent system of Camassa-Holm-Novikov equations
• Algebro-geometric solutions for the two-component Camassa-Holm Dym hierarchy
• Local well-posedness and global existence for a multi-component Novikov equation