Generalised Fourier transform for the Camassa-Holm hierarchy

TL;DR

This paper develops a generalized Fourier transform framework for the Camassa-Holm hierarchy by deriving a completeness relation for squared eigenfunctions, enabling a unified description of its integrals, hierarchy, and Hamiltonian structure.

Contribution

It introduces the completeness relation for squared solutions of the Camassa-Holm spectral problem, facilitating a comprehensive spectral analysis of the hierarchy.

Findings

Derived the completeness relation for squared solutions.

Unified description of integrals of motion and Hamiltonian structures.

Enabled spectral analysis using the generalized Fourier transform.

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.