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In this paper, we investigate explicit exact traveling wave solutions of the generalized (3+1)-dimensional KP equation

In this article, we investigate non-traveling wave solutions for the (2+1)-dimensional extended variable coefficients Bogoyavlenskii-Kadomtsev-Petviashvili equation with time-dependent coefficients (VC-BKP). Inspired by Prof. Shang, we apply the extended three-wave method and the generalized variable separation method to the investigated problem for the first time in this article. The technique is effective, easily applicable, and reliable in solving non-traveling wave solutions. We successfully obtain forty-four exact non-traveling solutions, including double periodic solutions, kinky breather wave solution, periodic cross-kink solution and some new exact non-traveling solutions obtained firstly in this paper. These results all have a tail which gives a prediction of physical phenomenon. Moreover, we discuss the arbitrary coefficients of solutions in the real, purely imaginary and complex domains, which greatly enriches the forms of solutions. The dynamic phenomena of four types of exact solutions are demonstrated by contour, 2D and 3D graphics, which help to show their physical interpretation.

<abstract><p>In this paper, we investigate non-traveling wave solutions of the (3+1)-dimensional variable coefficients Date-Jimbo-Kashiwara-Miwa (VC-DJKM) equation, which describes the real physical phenomena owing to the inhomogeneities of media. By combining the extended homoclinic test approach with variable separation method, we obtain abundant new exact non-traveling wave solutions of the (3+1)-dimensional VC-DJKM equation. These results with a parabolic tail or linear tail reveal the complex structure of the solutions for (3+1)-dimensional VC-DJKM equation. Moreover, the tail in these solutions maybe give a prediction of physical phenomenon. When arbitrary functions contained in these non-traveling wave solutions are taken as some special functions, we can get the kink-type solitons, singular solitary wave solutions, and periodic solitary wave solutions, and so on. As the special cases of our work, the corresponding results of (3+1)-dimensional DJKM equation, (2+1)-dimensional DJKM equation, (2+1)-dimensional VC-DJKM equation are also given.</p></abstract>