Bifurcation Geometry, Global Stability, and Nonlinear Nematicon Dynamics of the Generalized Hunter–Saxton Model

This study examines the generalized nonlinear Hunter–Saxton (HS) model: Φtx=ΦΦxx+γΦx2,γ≠0, that describes the evolution of spatial potential and angular velocity in the vector field of nematic liquid crystals. Closed-form nematicons are derived via the order reduction of the traveling wave ODE. The qualitative structures are analyzed for different values of the nonlinear parameter γ. The solutions are graphically depicted to discover rich nematicon geometries including parabolic, cuspon, kink, and singular wave structures. A comprehensive dynamic analysis of the reduced nonlinear ordinary system is performed using the phase plane method, which helps to reveal the non-isolated continuity of equilibrium and the role of singular manifolds in shaping the system’s sensitivity and stability. Bifurcation cases are investigated for distinct values of γ, and various transitions in trajectory geometry and semi-stability features are shown. The novelty appears in the comprehensive integrating of analytic and dynamic characterizations, through global phase and bifurcation analysis, of the generalized HS equation (HSE), which uncovers the control of nonlinear coefficient γ in governing the geometry and stability of the nematicons. Also, the analysis confirms the non-chaotic nature of the associated two-dimensional system, compatible with the Poincaré–Bendixson theorem.