The Two-Component Camassa-Holm Equations CH(2,1) and CH(2,2): First-Order Integrating Factors and Conservation Laws

TL;DR

This paper analyzes two multi-component generalizations of the Camassa-Holm equations, deriving first-order integrating factors and conservation laws to deepen understanding of their mathematical structure.

Contribution

It provides the first complete set of first-order integrating factors and conservation laws for the CH(2,1) and CH(2,2) systems, expanding their analytical framework.

Findings

Derived complete first-order integrating factors for CH(2,1) and CH(2,2)

Established conservation laws for both systems

Enhanced understanding of the systems' mathematical properties

Abstract

Recently, Holm and Ivanov, proposed and studied a class of multi-component generalisations of the Camassa-Holm equations [D D Holm and R I Ivanov, Multi-component generalizations of the CH equation: geometrical aspects, peakons and numerical examples, J. Phys A: Math. Theor. 43, 492001 (20pp), 2010]. We consider two of those systems, denoted by Holm and Ivanov by CH(2,1) and CH(2,2), and report a class of integrating factors and its corresponding conservation laws for these two systems. In particular, we obtain the complete sent of first-order integrating factors for the systems in Cauchy-Kovalevskaya form and evaluate the corresponding sets of conservation laws for CH(2,1) and CH(2,2).