Investigation on dynamical perspective of soliton solutions to the nonlinear integrable Akbota equation through a generalized analytical technique.

In this study, we investigate the nonlinear integrable Akbota equation (NLIAE), which is an important equation due to its energy-based nature and applications across various fields of science and engineering. Novel soliton solutions are investigated using the auxiliary equation method (AEM) applied to the NLIAE. The obtained soliton solutions exhibit unique, interesting, and diverse physical structures in the context of solitons, solitary, and traveling waves. To understand the physical implications of the derived solutions, a detailed computational analysis is performed, and several solutions are graphically represented in three formats contour, two-dimensional, and three-dimensional using numerical simulations with the aid of Mathematica. The graphical analysis reveals that the derived solutions possess novel physical structures of solitary waves and solitons, including periodic waves, kink waves, peakon bright and dark waves, singular bright and dark solitons, anti-kink waves, mixed kink-bright waves, and mixed anti-kink bright waves. The obtained results are expected to have potential applications in various domains of physical science and engineering, such as optics, ocean engineering, fluid mechanics, nonlinear dynamics, soliton and fractal theory, and other areas of nonlinear science. The findings demonstrate that the AEM is not only effective and straightforward but also an efficient tool for obtaining soliton solutions of various integrable equations, outperforming several existing methods. Moreover, this research presents novel solutions that reveal physical behaviors not previously reported in the literature.