The two-component <inline-formula><tex-math id="M1">\begin{document}$ \mu $\end{document}</tex-math></inline-formula>-Camassa–Holm system with peaked solutions

Abstract

This paper is mainly concerned with the classification of the general two-component $ \mu $-Camassa-Holm systems with quadratic nonlinearities. As a conclusion of such classification, a two-component $ \mu $-Camassa-Holm system admitting multi-peaked solutions and $ H^1 $-norm conservation law is found, which is a $ \mu $-version of the two-component modified Camassa-Holm system and can be derived from the semidirect-product Euler-Poincaré equations corresponding to a Lagrangian. The local well-posedness for solutions to the initial value problem associated with the two-component $ \mu $-Camassa-Holm system is established. And the precise blow-up scenario, wave breaking phenomena and blow-up rate for solutions of this problem are also investigated.