Painlev$$\acute{\mathrm{e}}$$ 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 Painlev $$\acute{\mathrm{e}}$$ integrable condition of the system. By virtue of the truncated Painlev $$\acute{\mathrm{e}}$$ expansion, we get an auto-Backlund transformation under certain Painlev $$\acute{\mathrm{e}}$$ integrable condition. Based on the bilinear form, we give a bilinear auto-Backlund transformation and a Lax pair under certain Painlev $$\acute{\mathrm{e}}$$ integrable condition. We obtain that a breather and kink waves propagate under certain Painlev $$\acute{\mathrm{e}}$$ 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 Painlev $$\acute{\mathrm{e}}$$ integrable conditions. During the propagation, the heights of the lump-periodic waves keep unchanged and height of the solitary wave periodically increases or decreases.