The semi-rational solutions of the (2+1)-dimensional cmKdV equations

The (2+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2+1$$\end{document})-D complex modified Korteweg–de Vries (cmKdV) equations are investigated with the aid of the Darboux transformation method. Through the limits λ2k-1→λ0=-a2+ci(k=1,…,m,m⩽n-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{2k-1}\rightarrow \lambda _0=-\frac{a}{2}+ci\,(k=1,\ldots ,m,\,m\leqslant n-1)$$\end{document}, the order-n semi-rational solutions are obtained. The order-2 semi-rational solutions and order-3 semi-rational solutions are analyzed in detail. By changing different parameters lj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_j$$\end{document}, different semi-rational solutions are deduced, including rogue wave interaction with the periodic wave or breather and lump interaction with the periodic wave or breather. The dynamical properties of these solutions are discussed, which indicates that these interactions are elastic collisions. In terms of application, these semi-rational solutions will be valuable in modeling physical problems.