New traveling-wave solutions for time-fractional generalized Hunter-Saxton model in nematic liquid crystals: A separation of variables approach

School of Data, Mathematical, and Statistical Sciences, University of Central Florida - Orlando, FL 32816, USA

Received: 16 October 2025
Accepted: 13 January 2026

Abstract

In this paper, a comprehensive analytical study of the time-fractional generalized Hunter-Saxton model is presented using a separation of variables approach. The model governs the propagation of orientation waves in massive nematic liquid crystals and exhibits intrinsic links to Einstein-Weyl geometric structures. Incorporating a fractional-order time derivative introduces temporal nonlocality, capturing memory-driven effects in the evolution of nonlinear reorientation fronts. An exact reduction to the traveling-wave frame yields closed-form families of solutions representing smooth, kink-type, and singular fronts that propagate at constant speed. For generic parameter regimes, algebraic profiles arise, while a resonant limit produces a smooth exponential front. The analysis further confirms the absence of real periodic traveling waves. Visualization of the exact solutions through three-dimensional surface, contour, and density plots reveals the influence of the fractional order on wave steepening, front morphology, and propagation dynamics, offering theoretical insights relevant to experimental exploration of reorientation phenomena in complex liquid-crystalline media.