Lie symmetries, stability, and chaotic dynamics of solitons in nematic liquid crystals with stochastic perturbation

This paper investigates the nonlinear and stochastic dynamics of solitons in nematic liquid crystals under the influence of multiplicative noise. Starting with a coupled nonlinear Schrödinger system and a reorientational equation, we apply a gauge transformation to remove stochastic perturbations modeled by Stratonovich calculus. Lie symmetry analysis is then employed to systematically reduce the resulting deterministic partial differential equations into ordinary differential equations. The dynamical system derived from the traveling-wave reduction is analyzed through phase portraits and Hamiltonian formulations, revealing various equilibrium structures. The presence of chaotic dynamics is confirmed by computing the Lyapunov exponent and visualizing phase transitions. We also derive explicit solitary and periodic wave solutions using elliptic and hyperbolic function representations. Our results demonstrate how Lie symmetries, combined with stochastic modeling, offer a powerful framework to understand and classify the complex behaviors of solitonic waves in nonlinear optical media.