Analytical investigation of solitary waves in a time-fractional Phi-four equation with dispersion

This study investigates the time-fractional Phi-four equation with an additional dispersion term, a model that captures anomalous diffusion and memory-dependent wave phenomena in physical systems. The equation extends classical frameworks by incorporating fractional temporal derivatives, which naturally arise when modeling processes exhibiting non-exponential relaxation and long-range temporal correlations. Such characteristics appear frequently in nonlinear optics, where ultrashort pulse propagation deviates from predictions made by integerorder models, and in condensed matter physics, where particle interactions display memory effects. We employ the Khater II method as our primary analytical technique, chosen for its demonstrated capacity to yield exact solutions with exceptional precision. This approach reveals several classes of solitary wave solutions, including kink, anti-kink, singular, and breather profiles, each representing distinct energy localization mechanisms. The balance between fractional dispersion and nonlinearity governs these structures, with the fractional derivative introducing temporal non-locality that fundamentally alters propagation characteristics compared to conventional models. To assess the physical relevance of these solutions, we perform stability analysis using Hamiltonian system characterization. The results confirm that solutions satisfy Lyapunov stability criteria across physically meaningful parameter regimes, indicating their robustness under small perturbations. Numerical verification supports our analytical findings, with computational errors remaining below machine precision levels. The fractional-order nature of the model introduces subtle but significant modifications to wave dynamics, particularly in how energy distributes spatially and evolves temporally. These findings advance our theoretical understanding of fractional dynamical systems while offering practical tools for modeling real-world phenomena where memory effects cannot be neglected. The systematic integration of fractional calculus with exact solution methods provides fresh insights into symmetry properties and coherent structure formation in systems governed by non-integer order evolution equations.