Stability of 2-soliton solutions in the modified Camassa-Holm equation

The modified Camassa-Holm equation with cubic nonlinearity is a completely integrable model, utilized to describe the unidirectional propagation of shallow-water waves. In this study, we show that the 2-soliton, when employed as a solution to the initial-value problem for the modified Camassa-Holm equation, is nonlinearly stable to perturbations within the Sobolev space H2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H<^>2$$\end{document} related to the momentum density. This stability is demonstrated through the conservation of crucial quantities in the Sobolev spaces H2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H<^>2$$\end{document}, H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H<^>{1}$$\end{document}, and L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>{1}$$\end{document} in terms of the momentum density.