Rogue Waves in the Three-Dimensional Kadomtsev–Petviashvili Equation

Breathers and rogue waves as exact solutions of the three-dimensional Kadomtsev–Petviashvili equation are obtained via the bilinear transformation method. The breathers in three dimensions possess different dynamics in different planes, such as growing and decaying periodic line waves in the (x,y), (x,z) and (y,t) planes. Rogue waves are localized in time, and are obtained theoretically as a long wave limit of breathers with indefinitely larger periods. It is shown that the rogue waves possess growing and decaying line profiles in the (x,y) or (x,z) plane, which arise from a constant background and then retreat back to the same background again.