Breather, rogue wave, lump, and interaction solutions of the $$(2+1)$$-dimensional generalized nonlocal Mel’nikov system

We investigate several kinds of localized wave solutions of the $$(2+1)$$ -dimensional generalized nonlocal Mel’nikov system by using the bilinear method and long-wave limit technique, which have found extensive applications in nonlinear wave theory, optics, and dynamical systems. We derive $$N$$ -soliton solutions by applying Hirota’s bilinear method and obtain breather solutions as well as breathers under periodic wave backgrounds through appropriate parameter restrictions. Taking the long-wave limit of soliton solutions yields the rational solutions, whereby kink-shaped rogue waves and lump solutions under a constant background are constructed. In particular, the dynamical characteristics of kink-shaped rogue waves are analyzed via asymptotic methods. Additionally, semi-rational solutions are obtained with the aid of the partial long-wave limit method, which includes (1) rogue waves and lumps under periodic wave backgrounds, and (2) interaction solutions between breathers and lumps. These analytical approaches enable multidimensional studies of nonlinear wave structures, providing a theoretical foundation for predicting novel wave behaviors in nonlocal nonlinear systems.