High-order doubly localized waves in (2+1)-dimensional Maccari system

In this work, we explore the doubly localized wave solutions for the (2+1)-dimensional Maccari system using Hirota’s bilinear method and Kadomtsev–Petviashvili (KP) hierarchy reduction method. These solutions illustrate the temporal evolution of a line rogue wave and a lump in the background of a breather or homoclinic orbit. The line rogue wave in these solutions is characterized by two endpoints, and is thus referred to as a line-segment rogue wave. The lump is also a doubly localized wave known as a rogue lump, which is also localized in both time and two spatial dimensions. These waves emerge from the breather waves or homoclinic orbits and remain unchanged for a short period of time, eventually merging into other breather waves or homoclinic orbits. Notably, these rogue lumps appear and vanish alongside the homoclinic orbits and play a significant role in understanding extreme events in physical scenarios.