Exploring new closed-form solitary waves of a nonlinear ionic transport model arising in microtubular mass flow and their global dynamical analysis

This work presents a comprehensive analytical and dynamical exploration of a nonlinear ionic currents microtubule model, which governs the propagation of ionic waves and electrical impulses along the cytoskeletal microtubule network. The governed model embodies the intrinsic nonlinear coupling between ionic conduction, capacitive effects, and external excitation, offering a realistic description of intracellular bioelectrical communication. By utilizing the (G′G2) method and the new extended algebraic equation (nEAE) approach, multiple classes of exact analytical solutions are obtained, including kink, anti-kink, singular, periodic, and mixed hyperbolic forms. These solutions capture a broad spectrum of nonlinear wave behaviors and provide valuable insight into signal modulation, energy transport, and localized excitations in microtubular assemblies. Beyond analytical construction, the system’s qualitative dynamics are systematically analyzed through equilibrium point classification, Jacobian stability, and bifurcation structures. The transition from ordered soliton states to chaotic oscillations is revealed via phase portraits, power spectra, Poincaré sections, and Lyapunov exponent analysis. The coexistence of positive and negative Lyapunov exponents confirms the emergence of deterministic chaos, while sensitivity and return-map analyses emphasize strong dependence on initial conditions. The combined results establish a deep correspondence between parameter variation and the onset of complex ionic oscillations within the microtubule channel. Physically, the derived wave solutions describe potential mechanisms of ionic pulse transmission, polarization reversal, and localized energy confinement in microtubule-based bioelectrical systems. The inclusion of chaotic and bifurcation features offers a new perspective on how biological nanostructures can amplify, regulate, or destabilize ionic signaling under nonlinear constraints. The proposed framework not only enriches the mathematical understanding of microtubule electrodynamics but also lays the groundwork for extending nonlinear analysis to other bio-inspired or nanotechnological transport systems.