Analysis of soliton behavior and overlap phenomena in the integrable beta-fractional Akbota equation with stability evaluation of equilibrium points

When analyzing the physical domain, beta derivatives are crucial for understanding wave dissemination across various nonlinear models. In this study, the integrable beta-fractional (1 + 1)-dimensional nonlinear Akbota equation is studied, which is important for understanding nonlinear processes in differential geometry, optics, and magnetism. To find soliton outcomes of this equation, we use the ϕ6-model expansion method. We also explore the stability of the equilibrium points with a potential theorem of the underlying nonlinear model. To demonstrate the dissemination appearances of the attained outcomes, the study includes physical illustrations in 3D, contour plots, and 2D plots, which are generated to observe how the parameters affect the solutions. Moreover, the overlapping phenomena of the obtained soliton outcomes are shown by scatter plots. The phase plane dynamics are analyzed by determining the Hamiltonian function. MATLAB, Python, and Maple software tools were used to run simulations. In addition to improving the mathematical tools for nonlinear systems, our work lays a strong basis for simulating complex wave events in advanced materials, fluid mechanics, and plasma physics. The lessons learnt about the nonlinear dynamic structures of these outcomes are unique and have never been explored before. The outcomes of this article offer insight into the complex dynamics of viscoelastic systems, advancing our understanding of the behavior of integrable Akbota equations and their soliton solutions.