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.