Shallow-water-wave studies on a (2 + 1)-dimensional Hirota–Satsuma–Ito system: X-type soliton, resonant Y-type soliton and hybrid solutions

•A (2 + 1)-dimensional Hirota–Satsuma–Ito system arising in the shallow water waves.•Some X- and Y-type solitons are determined via symbolic computation.•Hybrid solutions consisting of the resonance Y-type solitons and solitons/breathers/lumps are derived via symbolic computation.•Some nonlinear phenomena are depicted graphically. Water waves can be seen in the rivers, lakes, oceans, etc. A (2 + 1)-dimensional Hirota–Satsuma–Ito system, which arises in the shallow water waves, is investigated in this work. Based on the given N-soliton solutions, we develop certain X-type and resonant Y-type soliton solutions via the Hirota method and symbolic computation, where N is a positive integer. We also construct some hybrid solutions consisting of the resonant Y-type solitons, solitons, breathers and lumps. The graphics we present show that the hybrid solutions consisting of the resonant Y-type solitons and solitons/breathers/lumps, respectively, describe the interactions between the resonant Y-type solitons and solitons/breathers/lumps. The obtained results are dependent on the water-wave coefficient in that system.