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In the present paper, the potential Kadomtsev–Petviashvili equation and (3+1$3+1$)-dimensional potential-YTSF equation are investigated, which can be used to describe many mathematical and physical backgrounds, e.g., fluid dynamics and communications. Based on Hirota bilinear method, the bilinear equation for the (3+1$3+1$)-dimensional potential-YTSF equation is obtained by applying an appropriate dependent variable transformation. Then N-soliton solutions of nonlinear evolution equation are derived by the perturbation technique, and the periodic wave solutions for potential Kadomtsev–Petviashvili equation and (3+1$3+1$)-dimensional potential-YTSF equation are constructed by employing the Riemann theta function. Furthermore, the asymptotic properties of periodic wave solutions show that soliton solutions can be derived from periodic wave solutions.

Bilinear approach to soliton and periodic wave solutions of two nonlinear evolution equations of Mathematical Physics