Lie symmetry approach to the dynamical behavior and conservation laws of actin filament electrical models.

This research explores the dynamical properties and solutions of actin filaments, which serve as electrical conduits for ion transport along their lengths. Utilizing the Lie symmetry approach, we identify symmetry reductions that simplify the governing equation by lowering its dimensionality. This process leads to the formulation of a second-order differential equation, which, upon applying a Galilean transformation, is further converted into a system of first-order differential equations. Additionally, we investigate the bifurcation structure and sensitivity of the proposed dynamical system. When subjected to an external force, the system exhibits quasi-periodic behavior, which is detected using chaos analysis tools. Sensitivity analysis is also performed on the unperturbed system under varying initial conditions. Moreover, we establish the conservation laws associated with the equation and conduct a stability analysis of the model. Employing the tanh method, we derive exact solutions and visualize them through 3D and 2D graphical representations to gain deeper insights. These findings offer new perspectives on the studied equation and significantly contribute to the understanding of nonlinear wave dynamics.