Resonant collisions between lumps and periodic solitons in the Kadomtsev–Petviashvili I equation

Resonant collisions of lumps with periodic solitons of the Kadomtsev–Petviashvili I equation are investigated in detail. The usual lump is a stable weakly localized two-dimensional soliton, which keeps its shape and velocity in the course of the evolution from t → −∞ to t → +∞. However, the lumps would become localized in time as instantons, as a result of two types of resonant collisions with spatially periodic (quasi-1D) soliton chains. These are partly resonant and fully resonant collisions. In the former case, the lump does not exist at t → −∞, but it suddenly emerges from the periodic soliton chain, keeping its amplitude and velocity constant as t → +∞; or it exists as t → −∞ and merges into the periodic chain, disappearing at t → +∞. In the case of the fully resonant interaction, the lump is an instanton, which emerges from the periodic chain and then merges into another chain, keeping its identify for a short time. Thus, in the case of the fully resonant collisions, the lumps are completely localized in time as well as in two-dimensional space, and they are call rogue lumps.