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In this paper, we take the bidirectional sixth-order Sawada–Kotera equation as an instance and use a new limit approach to generate a multiple-pole solution and the degenerate of the breather wave from the <i>N</i>-order soliton solution. We show not only the substitution method, but also the specific mathematical expression of the double-pole, triple-pole, and the degenerate breather solution after the substitution. Meanwhile, we give the dynamic images and trajectories of the different multiple-pole solution. Moreover, we also acquire the interaction between two double-pole solutions and different nonlinear superposition solutions.