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 a four-wave mixing term are studied in this paper, which describe optical solitons in 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-shaped, multi-peak, and bright soliton solutions. Building upon single-periodic lump solutions, we analyze the dynamics of lump waves on both plane-wave and periodic backgrounds using the long-wave limit method. Moreover, we obtain the interaction solutions involving lumps, periodic lumps, and solitons. The interactions among two solitons, multiple lumps, and mixed waves are illustrated and analyzed. Comparative analysis reveals that these multi-lump solutions exhibit richer dynamical properties than conventional single-lump ones. These results contribute to a deeper understanding of nonlinear systems and may facilitate solving nonlinear problems in nature.