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The Cole–Hopf transformation is used to search for exact multiple wave solutions of Burgers’ equation. Particularly, solitary wave solutions and their interaction solutions are explicitly given, and a fission phenomenon is found, which is tough to be found by other traditional methods. Different solutions are presented with different combinations of elementary functions and a necessary condition for fission is proposed. Based on the bilinear formalism and with the aid of symbolic computation, we also determine two classes of rational solutions, rogue wave solutions and rogue–kink wave solutions, using various ansätze. To exhibit dynamics, we make diverse graphical analyses for the presented solutions, which show that the solutions are reliable in both mathematical physics and engineering.