The present article deals with multi-waves and breather wave solutions of the generalized Bogoyavlensky-Konopelchenko equation by virtue of the Hirota bilinear operator method and the semi-inverse variational principle. The obtained solutions for solving the current equation represent some localized waves including soliton, periodic, and cross-kink solutions in which have been investigated by the approach of the bilinear method. With certain parameter constraints in the multi-waves and breather, all cases of the periodic and cross-kink solutions can be captured from the one and two soliton(s). The obtained solutions are extended with numerical simulation to analyze graphically, which results into 1-and 2-soliton solutions and also periodic and cross-kink solutions profiles, that will be extensively used to report many attractive physical phenomena in the fields of acoustics, heat transfer, fluid dynamics, classical mechanics, and so on. KEYWORDS generalized Bogoyavlensky-Konopelchenko equation, Hirota bilinear operator method, multi-waves and breather, periodic and cross-kink solutions, semi-inverse variational principle, solitons MSC CLASSIFICATION

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