Study on the evolutionary dynamics of the paraxial wave equation applied to the liquid crystal model

In this study, we investigate the propagation dynamics of the M-truncated fractional paraxial wave equation arising in liquid crystal (LC). We construct exact analytical soliton solutions and analyze their structural and dynamical properties under fractional-order effects. By applying a traveling-wave transformation, the governing equation is reduced to an ordinary differential equation. A couple of efficient analytical schemes, such as the extended Jacobi elliptic function technique and the ([Formula: see text]-expansion method, are used to find a variety of exact solutions. The solutions include hyperbolic, trigonometric, rational and elliptic soliton forms. Several classes of wave structures, such as parabolic, peaked, periodic, cnoidal, asymptotic and breather-type solitons, are obtained. Particular parameter choices yield explicit solution families that clarify the effects of the model coefficients. Graphical representations in terms of modulus, real and imaginary parts illustrate the physical behavior and propagation characteristics of the obtained wave solutions. A comparative discussion between fractional-order and integer-order soliton solutions is also presented. The results demonstrate that the fractional paraxial model admits rich soliton dynamics and provides a flexible framework for describing nonlinear optical wave propagation in liquid-crystal media and related photonic systems.