A wave breaking criterion for a modified periodic two-component Camassa-Holm system

In this paper, a wave-breaking criterion of strong solutions is acquired in the Soblev space Hs(S)×Hs−1(S)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H^{s}(\mathbb{S})\times H^{s-1}(\mathbb{S})$\end{document} with s>32\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$s>\frac{3}{2}$\end{document} by employing the localization analysis in the transport equation theory, which is different from that of the two-component Camassa-Holm system.