Conformal and geometric properties of the Camassa-Holm hierarchy

TL;DR

This paper explores the conformal and geometric properties of the Camassa-Holm hierarchy, highlighting its integrable structure, spectral analysis, and solution construction via inverse scattering and generalized Fourier transforms.

Contribution

It introduces the conformal invariants of the CH hierarchy, details the spectral basis for inverse scattering, and constructs solutions for higher-dimensional generalizations.

Findings

Squared eigenfunctions form a complete basis for spectral analysis.

Explicit description of some CH hierarchy members using GFT.

Solutions for 2+1-dimensional CH generalizations via IST.

Abstract

Integrable equations with second order Lax pair like KdV and Camassa-Holm (CH) exhibit interesting conformal properties and can be written in terms of the so-called conformal invariants (Schwarz form). These properties for the CH hierarchy are discussed in this contribution. 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 (IST) for the Camassa-Holm hierarchy as a Generalised Fourier Transform (GFT). Using GFT we describe explicitly some members of the CH hierarchy, including integrable deformations for the CH equation. Also we show that solutions of some 2+1-dimensional generalizations of CH can be constructed via the IST for the CH hierarchy.