Nonlinear molecular deformations and solitons in a nematic liquid crystal

We studied nonlinear molecular deformations in a nematic liquid crystal with homeotropically aligned molecules and hard boundaries. As the basic dynamical equation for the director axis of the liquid crystal resembles the Landau-Lifshitz equation representing spin dynamics in a one dimensional classical continuum isotropic Heisenberg ferromagnetic spin chain, we invoke here the space curve formalism and the stereographic projection technique used in the case of the Heisenberg spin chain. Under space curve mapping, the director dynamics with elastic deformation is found to be governed by a perturbed nonlinear Schrödinger equation. A multiple-scale perturbation analysis brings out perturbed solitons to represent molecular deformations in the nematic liquid crystal. However, when a constant electric field is applied, the director dynamics is expressed under stereographic projection and the molecular deformations are found to be governed by periodic and localized static planar director configurations. A linear stability analysis on the static planar configurations shows that the system exhibits stable deformations.