(untitled) (European Journal of Pure and Applied Mathematics, 2025)
We use the Hirota bilinear technique to investigate wave solutions to the Heisenberg ferromagnetic- type Akbota problem arising in surface geometry. The equation is a crucial paradigm for examining and visualizing surface geometry and curve analysis. The equation is an integrable coupled model with soliton solutions. It is a key instrument for investigating nonlinear phenomena in differential geometry of curves and surfaces, magnetism, and optics. The solutions we explore are periodic lump waves, periodic cross-kink waves, homoclinic breather waves, M-shaped waves interacting with kink and rogue waves, and mixed waves. We derive exact solutions and carefully choose parameter values to generate a variety of three-dimensional graphs and the associated contour and density plots. The soliton phenomenon is explained by the physical structures and solutions discovered, which also replicate the dynamic characteristics of the traveling-wave deformation front that is created in the dispersive medium. It demonstrates the power, suitability, and potential applications of the Hirota bilinear technique in future research to identify unique solutions for a wide range of nonlinear model types found in engineering and physicalscience. These solutions are relevant in nonlinear fiber optics, magnetism and differential geometry.