Two-component generalizations of the Camassa-Holm equation

TL;DR

This paper classifies integrable two-component non-evolutionary PDE systems similar to the Camassa-Holm equation, identifies their bi-Hamiltonian structures, and constructs exact solutions and Lax pairs.

Contribution

It provides a comprehensive classification of such systems and their Hamiltonian structures, introducing new integrable models related to the Camassa-Holm equation.

Findings

Classified integrable two-component systems analogous to Camassa-Holm

Identified bi-Hamiltonian structures for these systems

Constructed exact solutions and Lax pairs

Abstract

A classification of integrable two-component systems of non-evolutionary partial differential equations that are analogous to the Camassa-Holm equation is carried out via the perturbative symmetry approach. Independently, a classification of compatible pairs of Hamiltonian operators is carried out, which leads to bi-Hamiltonian structures for the same systems of equations. Some exact solutions and Lax pairs are also constructed for the systems considered.