Qualitative analysis for a new generalized 2-component Camassa-Holm system

This paper considers the Cauchy problem for a 2-component Camassa-Holm system\begin{document}$ \begin{equation*} m_t = ( u m)_x+ u _xm- v m, \ \ n_t = ( u n)_x+ u _xn+ v n, \end{equation*} $\end{document}where \begin{document}$ n+m = \frac{1}{2}( u _{xx}-4 u ) $\end{document} , \begin{document}$ n-m = v _x $\end{document} , this model was proposed in [ 2 ] from a novel method to the Euler-Bernoulli Beam on the basis of an inhomogeneous matrix string problem. The local well-posedness in Sobolev spaces \begin{document}$ H^s(\mathbb{R})\times H^{s-1}(\mathbb{R}) $\end{document} with \begin{document}$ s>\frac{5}{2} $\end{document} of this system was investigated through the Kato's theory, then the blow-up criterion for this system was described by the technique on energy methods. Finally, we established the analyticity in both time and space variables of the solutions for this system with a given analytic initial data.