Generalised Fourier transform for the Camassa-Holm hierarchy

Abstract

The squared eigenfunctions of the spectral problem associated to the Camassa-Holm equation represent a complete basis of functions, which helps to describe the Inverse Scattering Transform for the Camassa-Holm hierarchy as a Generalised Fourier transform. The main result of this work is the derivation of the completeness relation for the squared solutions of the Camassa-Holm spectral problem. We show that all the fundamental properties of the Camassa-Holm equation such as the integrals of motion, the description of the equations of the whole hierarchy and their Hamiltonian structures can be naturally expressed making use of the completeness relation and the recursion operator, whose eigenfunctions are the squared solutions.

LaTeX, 40 pages, 1 Figure

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• Smooth and Peaked Solitons of the CH equation
• Equations of the Camassa-Holm Hierarchy
• Camassa-Holm cuspons, solitons and their interactions via the dressing method
• A dressing method for soliton solutions of the Camassa-Holm equation
• Generalised Fourier Transform and Perturbations to Soliton Equations
• Dynamical and spectral Dirac systems: response function and inverse problems
• Trace formulas and inverse spectral theory for generalized indefinite strings
• On the solution of the inverse problem for a class of canonical systems corresponding to matrix string equations
• On the (Non)-Integrability of KdV Hierarchy with Self-consistent Sources
• Local well-posedness and blow-up for a family of $U(1)$-invariant peakon equations