Azad Ali Sagher

The Akbota equation (AE), as a Heisenberg ferromagnetic-type equation, can be extremely valuable in the study of curve and surface geometry. In this study, we employ the well-known two analytical techniques, the modified Khater method and the new sub-equation approach, to construct the solitary wave solution of AE. Transform the partial differential equation into an ordinary differential equation using the wave transformation. The graphical visualization of select wave solutions is carried out using Wolfram Mathematica software. By utilizing appropriate parametric values across various wave velocities, this process unveils the intricate internal structures and provides a comprehensive understanding of wave behavior. The visual representations are rendered in 3D, 2D, and contour surfaces, capturing a range of solitonic phenomena. These include multiple kink solitons, flat kink, kink-peakon, kink solitons, and singular kink solitons, offering detailed insights into the complex dynamics of the system under study. Newly obtained soliton solutions are compared with available soliton solutions in the literature. The new results indicate that these obtained solutions can be a part of completing the family of solutions, and the considered methods are effective, simple, and easy to use. For qualitative assessment, convert the ordinary differential into a dynamical system by using the Galilean transformation to conduct the sensitivity visualization and bifurcation assessment along with phase portraits and chaos analysis of the considered model. Bifurcation analysis is crucial in soliton dynamics, as it influences the behavior and characteristics of solitons in various systems, with the results presented through phase portraits. Sensitivity visualization illustrates how parametric values affect the system’s behavior. The solutions obtained have broad applications in surface geometry and electromagnetism theory. The aim of this study is to enhance the understanding of complex nonlinear dynamics and their relevance in curve and surface geometry.