Dynamical visualization and propagation of soliton solutions of Akbota equation arising in surface geometry

This paper thoroughly investigates the integrable Akbota equation, a Heisenberg ferromagnet-type equation that plays a crucial role in exploring curve and surface geometry. The [Formula: see text]-model expansion method, known for its proficiency and reliability, is applied to generate generalized solitonic wave profiles, spanning a diverse range of soliton families. Prior to this study, there is no existing study in which this technique is utilized that ensured the existence of solution via development of conditions. On the other hand, this method is provided with the Jacobi elliptic function-based solutions with the limiting cases. The analytical strategy presented a notable advantage by specifying a constraint for each solution, ensuring its existence. Consequently, the solitonic wave structures exhibit various attributes, including the Jacobi elliptic function, periodicity, brightness, dark-brightness, singularity, exponential, trigonometry, and rational solitonic structures. These characteristics emerge under previously unexplored existence conditions. The results are visually depicted through 2D, 3D, and contour plots, providing a clear illustration of the behavioral responses to pulse propagation and allowing for the inference of fitting values for system parameters. This visualization offers valuable insights into the characteristics and dynamics of soliton solutions derived from the integrable Akbota equation.