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This paper mainly discussed the ( 3 + 1 )-dimensional Hirota–Satsuma–Ito-like equation. We obtain the novel lump solution, breather solution and the interaction between the lump solution and solitons and periodic wave. Firstly, the N-soliton solution is obtained based on Bell polynomial and Hirota bilinear. Secondly, on the basis of the multiple soliton solutions, the breather and interaction solution are obtained by using the complex conjugate construction method. We constructed two cases of breather interaction by adjusting appropriate parameters and studied the dynamic behavior of 2 breathers in detail combined with three-dimensional plots and density plots. Finally, by constructing a new positive quadratic function method, we obtained lump solutions and interaction solutions with kink soliton, line soliton, and periodic wave. The interaction between a lump and 1-stripe soliton is especially discussed. These solutions and properties are useful to explain the physical phenomena described by the ( 3 + 1 )-dimensional HSIl equation. So far, the results obtained in this paper have not been mentioned in previously published research. Our research methods and analytical results enrich the dynamics of the ( 3 + 1 )-dimensional HSIl equation. [ABSTRACT FROM AUTHOR]