Dynamical analysis and bifurcations in a fractional integrable equation

Abstract

This paper investigates the dynamics of the nonlinear (1+1)-dimensional integrable beta-fractional Akbota equation, a Heisenberg ferromagnet-type model essential for understanding curve analysis and surface geometry. Characterized as a system of coupled differential equations with soliton solutions, this equation plays a crucial role in studying nonlinear processes in differential geometry, magnetism, and optics. Exact soliton wave solutions, including periodic, dark, trigonometric, Jacobi elliptic, Weierstrass elliptic function solutions, hyperbolic, rational, and solitary waves, are derived using the novel sub-ODE method and the unified method. These analytical solutions enhance the theoretical framework, allowing for the identification of broader patterns and relationships in nonlinear wave phenomena. To illustrate the propagation characteristics of the obtained soliton solutions, the study includes physical representations such as $2D$ and $3D$ visualizations. Additionally, a qualitative analysis of the dynamical system is conducted, investigating chaotic behavior through bifurcation theory. Notably, this study is the first to apply these soliton solution methods to the fractional Akbota equation. Furthermore, the equation is transformed into a planar dynamical system via the Galilean transformation, and its sensitivity performance is assessed. A detailed sensitivity analysis, implemented using the Runge–Kutta numerical integration scheme, reveals that bounded deviations in solution trajectories arise from perturbations in initial state vectors. This systematic investigation establishes the structural stability characteristics of the system within the examined parameter domain, demonstrating that the solution manifolds maintain robust stability under infinitesimal variations in initial conditions.