We analyze in detail the interactions of two-dimensional solitary waves called lumps and onedimensional line solitons within the framework of the Kadomtsev–Petviashvili equation describing wave processes in media with positive dispersion. We show that line solitons can emit or absorb lumps or periodic chains of lumps, as well as interact with each other by means of lumps. Within a certain time interval, lumps or lump chains can emerge between two line solitons and then disappear due to absorption by one of the solitons. This phenomenon resembles the appearance of rogue waves in the oceans. The results obtained are graphically illustrated and can be applicable to the description of the physical processes occurring in plasmas, fluids, solids, nonlinear optical media, etc.