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In the present paper, we investigate a-dimensional nonlinear evolution equation, which can be used to describe the shallow-water waves. Via the Kadomtsev–Petviashvili hierarchy reduction, we derive the breather, semi-rational and rational solutions in terms of Gramian. The breathers are dark breathers. Taking the long wave limit in the breather solutions, we derive the semi-rational solutions describing the interactions between the lumps and breathers, and the rational solutions which give birth to the lumps are also derived. Interactions between the lumps and breathers are elastic. Furthermore, we graphically present the interactions among the dark lumps. We find that the interactions among the three lumps are elastic.

Breathers, interactions between breathers and lumps for a (3 + 1)-dimensional nonlinear evolution equation for the shallow-water waves