Equations of the Camassa-Holm Hierarchy

TL;DR

This paper explores the mathematical structure of the Camassa-Holm hierarchy using squared eigenfunctions, providing a generalized Fourier transform approach to describe its properties, solutions, and integrable deformations.

Contribution

It introduces a GFT framework based on squared eigenfunctions for the CH hierarchy, enabling explicit descriptions of hierarchy members and solutions.

Findings

Complete basis of squared eigenfunctions for the spectral problem

Explicit description of some CH hierarchy members using GFT

Construction of solutions, including peakons, for higher-dimensional CH equations

Abstract

The squared eigenfunctions of the spectral problem associated with the Camassa-Holm (CH) equation represent a complete basis of functions, which helps to describe the inverse scattering transform for the CH hierarchy as a generalized Fourier transform (GFT). All the fundamental properties of the CH equation, such as the integrals of motion, the description of the equations of the whole hierarchy, and their Hamiltonian structures, can be naturally expressed using the completeness relation and the recursion operator, whose eigenfunctions are the squared solutions. Using the GFT, we explicitly describe some members of the CH hierarchy, including integrable deformations for the CH equation. We also show that solutions of some $(1+2)$ - dimensional members of the CH hierarchy can be constructed using results for the inverse scattering transform for the CH equation. We give an example of the peakon solution of one such equation.