Painlevé analysis, auto-Bäcklund transformation and new exact solutions of (2+1) and (3+1)-dimensional extended Sakovich equation with time dependent variable coefficients in ocean physics

This article considers time-dependent variable coefficients $(2+1)$ and $(3+1)$-dimensional extended Sakovich equation. Painlevé analysis and auto-Bäcklund transformation methods are used to examine both the considered equations. Painlevé analysis is appeared to test the integrability while an auto-Bäcklund transformation method is being presented to derive new analytic soliton solution families for both the considered equations. Two new family of exact analytical solutions are being obtained successfully for each of the considered equations. The soliton solutions in the form of rational and exponential functions are being depicted. The results are also expressed graphically to illustrate the potential and physical behaviour of both equations. Both the considered equations have applications in ocean wave theory as they depict new solitary wave soliton solutions by 3D and 2D graphs.