A Novel Higher-Order Dispersive Extension of the Generalized Hunter–Saxton Model: Traveling-wave Solutions, Time-Fractional Impacts, and Qualitative Analysis

This study introduces and analyzes a new higher-order extension of the generalized Hunter–Saxton equation, which will be known by the fourth–order dispersive Hunter–Saxton equation. Unlike the classical version, novel fourth–order dispersive extension allows traveling–wave frameworks, that aren’t applicable before. Closed–form soliton solutions, including rational (kink-type) and periodic, are derived via the extended auxiliary equation method. Existence of these solutions is shown through free parameters. To quantitatively assess the effects of time–fractional , modified Riemann-Liouville, β, and M-truncated derivatives are considered. The corresponding traveling–wave reductions are illustrated to track these effects. Finally, the reduced dynamical system is examined numerically, revealing a remarkable sensitivity to starting data and bounded transverse oscillations. Also, the stability analysis shows a non-isolated equilibrium that induces the oscillating between neutral divergence, oscillatory centered, and hyperbolic saddle system. These results illustrate how both fourth-order dispersion and fractional-time structures enrich the dynamics of Hunter–Saxton model.