Painleve´\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\acute{\mathrm{e}}$$\end{document} integrable condition, auto-Bäcklund transformations, Lax pair, breather, lump-periodic-wave and kink-wave solutions of a (3+1)-dimensional Hirota–Satsuma–Ito-like system for the shallow water waves

In this paper, we investigate a (3+1)-dimensional Hirota–Satsuma–Ito-like system for the shallow water waves. We obtain a Painleve´\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\acute{\mathrm{e}}$$\end{document} integrable condition of the system. By virtue of the truncated Painleve´\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\acute{\mathrm{e}}$$\end{document} expansion, we get an auto-Bäcklund transformation under certain Painleve´\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\acute{\mathrm{e}}$$\end{document} integrable condition. Based on the bilinear form, we give a bilinear auto-Bäcklund transformation and a Lax pair under certain Painleve´\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\acute{\mathrm{e}}$$\end{document} integrable condition. We obtain that a breather and kink waves propagate under certain Painleve´\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\acute{\mathrm{e}}$$\end{document} integrable condition. The breather has a peak and a trough and the height of the kink wave periodically increases or decreases during the propagation. Furthermore, we get the lump-periodic-wave and solitary-wave solutions and observe that the lump-periodic and solitary waves propagate under certain Painleve´\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\acute{\mathrm{e}}$$\end{document} integrable conditions. During the propagation, the heights of the lump-periodic waves keep unchanged and height of the solitary wave periodically increases or decreases.