Lump and Mixed Rogue‐Soliton Solutions of the (2 + 1)‐Dimensional Mel’nikov System

Abstract

Lump wave and line rogue wave of the (2 + 1)-dimensional Mel’nikov system are derived by taking the ansatz as the rational function. By combining a rational function and different exponential functions, mixed solutions between the lump and soliton are derived. These solutions describe the interaction phenomena of the lump-bright soliton with fission and fusion, the half-line rogue wave with a bright soliton, and a rogue wave excited from the bright soliton pair, respectively. Some special concrete interaction solutions are depicted in both analytical and graphical ways.

Resonant collisions among two-dimensional localized waves in the Mel’nikov equation

“…The general bright and dark solitary wave solutions of the Mel'nikov equation have been reported by Hase et al [19] and Lakshmanan et al [20] have investigated the exponentially localized solutions. The rational localized solutions have been calculated by Mu et al [21], Deng et al [22] have used the ansatz technique to derive lump and mixed rogue wave solutions, Sun et al [23] have discussed the semi-rational solutions that describe the resonant collision between lumps and solitons, Zhang et al [24] have reported the so-called hybrid solutions, and Rao et al [25] have investigated two families of semi-rational solutions of Mel'nikov equation using the KP-hierarchy reduction method and the Hirota direct method combined with a perturbative expansion technique. Recently, twodimensional coherent structures of lump-soliton solutions of the Mel'nikov equation have been investigated by Liu et al [26].…”