Soliton dynamics, overlapping phenomena, multistability, and chaotic nature of a nonlinear model in relativistic wave mechanics

This manuscript explores exact soliton solutions of the fractional (2 + 1)-dimensional Heisenberg ferromagnetic spin chain (HFSC) equation using advanced mathematical methods relevant to modeling nonlinear wave propagation in a complex physical medium. The key emphasis of this manuscript is to use generalized conformable derivatives (GCD) and the [Formula: see text] model expansion techniques to obtain precise soliton solutions for the HFSC equation. We used the wave transformation to change the studied partial differential equation (PDE) into an ordinary differential equation (ODE). Then we obtain diverse phenomena, such as anti-compacton soliton, compacton soliton, bright soliton, periodic wave, and kink-type shockwave, with overlapping features between the two solutions. We also applied several chaotic tools, including the bifurcation diagram, return map, power spectrum, basins of attraction, fractional dimension, recurrence plot, and multistability analysis. Simulations were performed using MATLAB and Maple. Additionally, soliton outcomes for the (2 + 1) dimensional HFSC model with conformable derivatives (CD) have been displayed in three dimensions (3D), two dimensions (2D), and contour plots. Our work establishes a solid foundation and a unified framework for investigating and understanding nonlinear fractional systems underlying complex wave phenomena in plasma physics, advanced materials, and fluid mechanics, while also improving the mathematical apparatus for nonlinear systems. People use the outcomes of solutions in fluids, optics, plasma physics, Bose-Einstein condensates, traffic flow, population dynamics, biophysics, shallow-water waves, and many other domains.