Higher-order lump and semi-rational solutions for the $$(2+1)$$-dimensional nonlocal Mel’nikov equation

This paper systematically investigates the high-order lump solutions and semi-rational solutions of the $$(2+1)$$ -dimensional nonlocal Mel’nikov equation satisfying the PT -symmetric condition. By employing Hirota’s bilinear method and the Kadomtsev–Petviashvili (KP) hierarchy reduction technique, the general determinant form of these solutions expressed in terms of Schur polynomials and exponential functions is established. Based on this framework, we classify first-order lump solutions into three types: bright lump, four-petaled lump, and dark lump, and further extend this classification to higher-order cases. The semi-rational solutions in which lumps coexist with dark solitons and breathers are constructed. Furthermore, we systematically analyze their dynamical behaviors, and elucidate the controlling role of parameter aₖ on the position of these lumps. This study not only extends the structure of the analytical solutions for the nonlocal Mel’nikov equation but also provides a theoretical foundation for understanding interaction mechanisms of nonlinear waves in the PT -symmetric systems. These solutions have significant implications for fields such as optics, plasma physics, and fluid mechanics.