Dynamics of general higher-order rogue waves in the two-component nonlinear Schrödinger equation coupled to the Boussinesq equation

Abstract

We report three families of general higher-order rogue wave solutions to the two-component nonlinear Schrödinger equation coupled to the Boussinesq (2NLS–Boussinesq) equation via the bilinear Kadomtsev–Petviashvili hierarchy reduction method, which governs two short waves (SWs) interacting with a common long wave (LW). In the first solution family, the fundamental rogue waves in the SW components admit three different states: bright, dark, and four-petaled ones, whereas they are always bright state in the LW component. The higher-order rogue waves are superpositions of N(N+1)2 bounded fundamental rogue waves. The second solution family can be regarded as an extension of the first family, which is a superposition of the N1th-order and N2th-order bounded rogue waves in the first solution family. These N1th-order and N2th-order bounded rogue waves are in different states, thus the rogue waves of composite states can be found, including bright–dark rogue waves, bright-four-petaled rogue waves, and dark-four-petaled rogue waves. The third family contains the degenerated solutions of the first family, and consists of [N̂12+N̂22−N̂1(N̂2−1)] bounded fundamental rogue waves. Here N,N1,N2 are positive integers, while N̂1 and N̂2 are nonnegative integers. Some degenerated rogue waves are exhibited, such as the special cases consisting of two, four, or five bounded fundamental rogue waves.