Harvesting Mixed, Homoclinic Breather, M-Shaped, and Other Wave Profiles of the Heisenberg Ferromagnet-Type Akbota Equation

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 toidentify 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.