The present article deals with multi-waves and breathers solution of the (2+1)-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation under the Hirota bilinear operator method. The obtained solutions for solving the current equation represent some localized waves including soliton, solitary wave solutions, periodic and cross-kink solutions in which have been investigated by the approach of the bilinear method. Mainly, by choosing specific parameter constraints in the multi-waves and breathers, all cases the periodic and cross-kink solutions can be captured from the 1- and 2-soliton. The obtained solutions are extended with numerical simulation to analyze graphically, which results in 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. We have shown that the assigned method is further general, efficient, straightforward, and powerful and can be exerted to establish exact solutions of diverse kinds of fractional equations originated in mathematical physics and engineering. We have depicted the figures of the evaluated solutions in order to interpret the physical phenomena.