Summary: The present article deals with \(M\)-soliton solution and \(N\)-soliton solution of the \((2+1)\)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation by virtue of Hirota bilinear operator method. The obtained solutions for solving the current equation represent some localized waves including soliton, breather, lump, and their interactions, which have been investigated by the approach of the long-wave limit. Mainly, by choosing the specific parameter constraints in the \(M\)-soliton and \(N\)-soliton solutions, all cases of the one breather or one lump can be captured from the two, three, four, and five solitons. In addition, the performances of the mentioned technique, namely, the Hirota bilinear technique, are substantially powerful and absolutely reliable to search for new explicit solutions of nonlinear models. Meanwhile, the obtained solutions are extended with numerical simulation to analyze graphically, which results in localized waves and their interaction from the two-, three-, four-, and five-soliton solutions profiles. They will be extensively used to report many attractive physical phenomena in the fields of acoustics, heat transfer, fluid dynamics, classical mechanics, and so on.