Exploring the wave's structures to the nonlinear coupled system arising in surface geometry

This manuscript deals with the Heisenberg ferromagnet-type integrable Akbota equation (AE), which refers to a set of differential equations that are integrable and linked together, and they possess solitary waves. AE is a basic gear for investigating nonlinear dynamics in the fields of optics, magnetism, and differential geometry of curves and surfaces. It is extensively used to represent optical solitons in nonlinear optical fibers, which are crucial for fiber-optic communication owing to their capacity to maintain form across considerable distances. The dynamical behavior of AE is explored by constructing accurate closed-form traveling wave solutions. For this purpose, the Kumar-Malik method, the new Kudryashov method, and the Riccati equation method are utilized. The resulting solutions consist of trigonometric, hyperbolic, and rational functions. By employing these methodologies, precise analytical remedies for soliton waves are derived, which include kink, bright, and dark solitons. To get a better understanding of the physical aspects of these solutions, we depict them via several visual representations. 3D-surface graphs, 2D-line graphs, and contour and density plots, in addition to theoretical derivations.