Smooth and Peaked Solitons of the CH equation

Abstract

The relations between smooth and peaked soliton solutions are reviewed for the Camassa-Holm (CH) shallow water wave equation in one spatial dimension. The canonical Hamiltonian formulation of the CH equation in action-angle variables is expressed for solitons by using the scattering data for its associated isospectral eigenvalue problem, rephrased as a Riemann-Hilbert problem. The momentum map from the action-angle scattering variables $T^{\ast }({\mathbb{T}}^N)$ to the flow momentum (${\mathfrak{X}}^{\ast }$) provides the Eulerian representation of the $N$-soliton solution of CH in terms of the scattering data and squared eigenfunctions of its isospectral eigenvalue problem. The dispersionless limit of the CH equation and its resulting peakon solutions are examined by using an asymptotic expansion in the dispersion parameter. The peakon solutions of the dispersionless CH equation in one dimension are shown to generalize in higher dimensions to peakon wave-front solutions of the EPDiff equation whose associated momentum is supported on smoothly embedded subspaces. The Eulerian representations of the singular solutions of both CH and EPDiff are given by the (cotangent-lift) momentum maps arising from the left action of the diffeomorphisms on smoothly embedded subspaces.

First version -- comments welcome! Submitted to JPhysA

References in corpus (5)

• Inverse Scattering Transform for the Camassa-Holm equation
• On asymptotically equivalent shallow water wave equations
• Generalised Fourier transform for the Camassa-Holm hierarchy
• Poisson structure and Action-Angle variables for the Camassa-Holm equation
• Momentum Maps and Measure-valued Solutions (Peakons, Filaments and Sheets) for the EPDiff Equation

Cited by in corpus (11)

• Two-component {CH} system: Inverse Scattering, Peakons and Geometry
• Multi-component generalizations of the {CH} equation: Geometrical Aspects, Peakons and Numerical Examples
• Bäcklund transformation and smooth multisoliton solutions for a modified Camassa-Holm equation with cubic nonlinearity
• Waltzing peakons and compacton pairs in a cross-coupled Camassa-Holm equation
• A view of the peakon world through the lens of approximation theory
• Liouville Correspondences between Integrable Hierarchies
• Multisoliton solutions of the two-component Camassa-Holm system and their reductions
• Multicomponent Fokas-Lenells equations on Hermitian symmetric spaces
• New variational and multisymplectic formulations of the Euler-Poincaré equation on the Virasoro-Bott group using the inverse map
• Integrable negative flows of the Heisenberg ferromagnet equation hierarchy
• Geometric numerical integrators for Hunter-Saxton-like equations