On integrability of the Camassa-Holm equation and its invariants. A geometrical approach

TL;DR

This paper investigates the integrability of the Camassa-Holm equation using a geometric framework, detailing its Hamiltonian, symplectic structures, and symmetries, both in scalar and 2x2-system forms.

Contribution

It provides a comprehensive geometric analysis of the Camassa-Holm equation, including new insights into its Hamiltonian, symplectic structures, and conservation laws.

Findings

Identification of Hamiltonian and symplectic structures

Derivation of recursion operators and symmetries

Construction of infinite series of conservation laws

Abstract

Using geometrical approach exposed in arXiv:math/0304245 and arXiv:nlin/0511012, we explore the Camassa-Holm equation (both in its initial scalar form, and in the form of 2x2-system). We describe Hamiltonian and symplectic structures, recursion operators and infinite series of symmetries and conservation laws (local and nonlocal).