Fractional Solutions and Quasi-Periodic Dynamics of the M-Fractional Weakly Nonlinear Dispersive Water Waves Model: A Bifurcation Perspective

In this paper, we study the time-space truncated M-fractional model of shallow water waves in a weakly nonlinear dispersive media that characterizes the nano-solitons of ionic wave propagation along microtubules in living cells. A fractional wave transformation is applied, reducing the model to a third-order differential equation formulated as a conservative Hamiltonian system. The stability of the equilibrium points is analyzed, and the corresponding phase portraits are constructed, providing valuable insights into the expected types of solutions. Utilizing the dynamical systems approach, a variety of predicted exact fractional solutions are successfully derived, including solitary, periodic and unbounded singular solutions. One of the most notable features of this approach is its ability to identify the real propagation regions of the desired waves from both physical and mathematical perspectives. The impacts of the fractional order and gravitational force variations on the solution profiles are systematically analyzed and graphically illustrated. Moreover, the quasi-periodic dynamics and chaotic behavior of the model are explored.