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.

中文翻译：

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两部分 \ begin {document} $ \ mu $ \ end {document}-Camassa–Holm系统，带峰值解决方案

本文主要涉及具有二次非线性的一般两元系统\ \ mu $ -Camassa-Holm系统的分类。作为这种分类的结论，找到了一个包含多峰解的两部分$ \ mu $ -Camassa-Holm系统和$ H ^ 1 $范数守恒定律，这是两个变量的$ \ mu $版本分量的改进的Camassa-Holm系统，并且可以从对应于Lagrangian的半直接乘积Euler-Poincaré方程导出。建立了与两分量$ \ mu $ -Camassa-Holm系统相关的初始值问题的解的局部适定性。并研究了精确的爆炸场景，波浪破裂现象和解决该问题的爆炸率。