Periodic lump, soliton and some mixed solutions of the (2+1)-dimensional generalized coupled nonlinear Schrödinger equations

The (2+1)-dimensional generalized coupled nonlinear Schrödinger equations with the four-wave mixing term is studied in this paper, which describes the optical solitons in the birefringent fibers. Utilizing the Hirota bilinear method, we systematically construct single and double periodic lump solutions. To provide a detailed insight into the dynamic behavior of the nonlinear waves, we explore diverse mixed solutions including bright-dark, W-shape, multi-peak and bright soliton solutions. Building upon single periodic lump solutions, we analyze the dynamics of lump waves on both the plane wave and periodic backgrounds by long wave limit method. Besides, we get the interaction solutions of lump, periodic lump and solitons. Moreover, the interaction of two solitons, multiple lumps and mixed waves are studied through illustrations. Comparative analysis reveals these multi-lump solutions exhibit richer dynamical properties compared to conventional one lump. Such results can help people understand nonlinear systems and solve nonlinear problems in nature.