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

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).

💡 Research Summary

The paper focuses on two multi‑component generalisations of the Camassa‑Holm (CH) equation introduced by Holm and Ivanov in 2010, namely the systems denoted CH(2,1) and CH(2,2). Both models extend the classical CH equation by incorporating additional dependent variables (e.g., a second velocity component, a density field, and auxiliary fields) while preserving the characteristic nonlinear, non‑local structure that gives rise to peakon solutions. The authors’ primary goal is to determine all first‑order integrating factors for these systems when they are written in Cauchy‑Kovalevskaya (CK) form, and to use these factors to construct the associated conservation laws.

The analysis proceeds in several well‑defined stages. First, each system is rewritten as a first‑order quasilinear system \