New similarity reductions and exact solutions of the $(2+1)$-dimensional extended Bogoyavlenskii–Kadomtsev–Petviashvili equation

The CK direct method is employed to investigate exact solutions of the $(2+1)$-dimensional extended Bogoyavlenskii–Kadomtsev–Petviashvili equation, which usually describes the propagation of nonlinear waves in various fields, such as fluid dynamics and plasma physics. It is extremely challenging to derive exact solutions for the eBKP equation. We have found that at present there is no research in the scientific literature on the application of the CK method to the eBKP equation due to tedious and complex calculations and inherent difficulty in determining explicit expressions for $\beta$ and $z$. To address these limitations, we adopt a separation-of-equations approach to find concrete expressions for $\beta$ and $z$. Through an extensive series of complex calculations, we successfully obtain new similarity reductions and new exact solutions for the eBKP equation, including Painlevé-type reductions, Weierstrass elliptic function solutions, and rational solutions that have not been reported in prior studies. Solutions of the eBKP equation can successfully degenerate into those of the BKP equation. From a physical perspective, through the analysis of the new solutions to the BKP equation, we find that as $t$ gradually increases, wave BKP solutions develop progressive instability and exhibit a tendency toward collapse. We find that introducing extended dispersion terms in the BKP equation enhances the amplitude of wave solutions and induces a tilting effect on wave propagation along the crest line.