Analysis of the dynamical behaviors for the generalized Bogoyavlvensky–Konopelchenko equation and its analytical solutions occurring in mathematical physics

In the domains of fluid mechanics, hydrodynamics, and marine engineering, Bogoyavlensky–Konopelchenko equations are of great interest to mathematicians and physicists as a means of illuminating the diverse dynamics of non-linear wave events. In this study, to pique readers' interest, we investigate the soliton solutions of a dynamical model, which is the mathematical physics equivalent of the (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation (GBKE). Utilizing the improved modified extended tanh-function scheme (IMETFS), we generate several innovative solutions. Utilizing the previously described approach, we find new types of solutions that have never been found before to demonstrate their originality for the problem at hand, such as dark, singular soliton, exponential, hyperbolic, singular periodic, Jacobi elliptic function (JEF), and rational solutions. The results show that the computational procedures are clear, informed, and effective. By integrating them with representational calculations, they may be used for more intricate phenomena. The efficacy of our method indicates that it may be utilized to tackle other non-linear challenges in many domains, particularly in soliton theory, since the examined model appears in many applications. Utilizing the computer algebra system, Wolfram Mathematica®, the propagation of the well-furnished results is visualized through contour plots, 2D and 3D visualizations for different values of the required free parameters. All of the research's conclusions are necessary to comprehend the behavior and physical significance of the examined equation, highlighting how crucial it is to examine various non-linear wave phenomena in the field of engineering mathematics and physical sciences.