# Nematic Dispersive Shock Waves from Nonlocal to Local
## 2 Nematic Equations  
Let us consider the propagation of a linearly, extra-ordinarily polarized, coherent light beam of wavenumber \(k_{0}\), wavelength \(\lambda_{0}=2\pi/k_{0}\), through a planar cell filled with nematic liquid crystals. The optical beam is assumed to propagate down the cell along in the \(Z\) direction, with its electric field \(E\) polarized in the \(Y\) direction. The coordinate \(X\) then completes the coordinate system. Nematic liquid crystals are a uniaxial medium consisting of elongated molecules, with the long axis termed the molecular director. The refractive index of the medium is \(n_{\parallel}\) for optical beams polarized along the molecular director and \(n_{\perp}\) for fields polarized orthogonal to the director. A fundamental property of nematic liquid crystals is the so-called Freedericksz threshold, whereby a minimum optical power is needed to rotate the nematic molecules, thus changing the refractive index of the medium [1]. However, high optical powers lead to heating of the nematic medium, which can cause the loss of the nematic state if the temperature change is high enough [1,6]. One method to overcome this is to pre-tilt the nematic molecules at an angle \(\theta_{0}\) with respect to the \(Z\)-direction upon the application of an external static electric field \(E_{LF}\), so that milli-Watt power beams can rotate the nematic molecules [6]. Let us denote the optically induced rotation of the nematic by \(\phi\), so that, in the presence of an optical beam, the total angle of the nematic director to the \(Z\) direction is \(\theta=\theta_{0}+\phi\). This configuration of the nematic cell is illustrated in Figure 1. The dimensional equations governing the propagation of the optical beam in the nematic cell are then of the following form:  
\[2 ik_{0}n_{e}\frac{\partial E}{\partial Z}+\
abla^{2}E+k_{0}^{2}\Big{[}n_{ \perp}^{2}\cos^{2}\theta+n_{\parallel}^{2}\sin^{2}\theta-n_{\perp}^{2}\cos^{ 2}\theta_{0}-n_{\parallel}^{2}\sin^{2}\theta_{0}\Big{]}E=0, \tag{1}\]  
for the electric field of the beam, and  
\[K\
abla^{2}\phi+\left[\frac{1}{4}\epsilon_{0}\Delta\epsilon|E|^{2}+\frac{1}{2} \Delta\epsilon_{LF}E_{LF}^{2}\right]\sin 2(\theta_{0}+\phi)=0, \tag{2}\]for the nematic response [2; 3; 4; 5]. Here, the extraordinary refractive index of the nematic is:  
\[n_{e}^{2}=\frac{n_{\perp}^{2}n_{\parallel}^{2}}{n_{\parallel}^{2}\cos^{2}\theta+ n_{\perp}^{2}\sin^{2}\theta}. \tag{3}\]  
In the above equations, \(\Delta\epsilon=n_{\parallel}^{2}-n_{\perp}^{2}\) is the optical anisotropy, \(\Delta\epsilon_{LF}\) is the low-frequency dielectric anisotropy, and \(\epsilon_{0}\) is the electrical permittivity of free space. In addition, the constant \(K\) is the elastic medium constant in the one constant approximation for which the elastic constants of bend, twist, and splay are taken equal [1; 2]. The nematic Equations (1) and (2) are highly nonlinear and difficult to analyze. However, for milli-Watt power beams, the optically induced rotation \(\phi\) is small, \(|\phi|\ll\theta_{0}\), so that these equations can be expanded in Taylor series around \(\theta_{0}\). In addition, these equations can be put in dimensionless form using typical scales \(L_{Z}\) down the cell and \(W\) transverse to the down cell direction, as well as a typical scale \(A_{b}\) for the electric field of the optical beam, so that  
\[Z=L_{z}z,\quad X=Wx,\quad Y=Wy,\quad E=A_{b}u. \tag{4}\]  
Here, \((x,y,z)\) is the non-dimensional coordinate system, and \(u\) is the non-dimensional electric field of the optical beam.