# Nematic Dispersive Shock Waves from Nonlocal to Local
## 2 Nematic Equations
The electric field scale is obtained by assuming that the input optical beam is a Gaussian beam of power \(P_{b}\), amplitude \(A_{b}\), and width \(W_{b}\), so that  
\[A_{b}^{2}=\frac{2P_{b}}{\pi\Gamma W_{b}^{2}},\quad\Gamma=\frac{1}{2}\epsilon_{ 0}cn_{e},\quad n_{e}^{2}=\frac{n_{\parallel}^{2}n_{\perp}^{2}}{n_{\parallel} ^{2}\cos^{2}\theta_{0}+n_{\perp}^{2}\sin^{2}\theta_{0}}. \tag{5}\]  
Substituting these into the nematic Equations (1) and (2), and expanding in Taylor series for small \(|\phi|\), we find [5; 39] that suitable scalings are  
\[L_{Z}=\frac{4n_{e}}{\Delta\epsilon k_{0}\sin 2\theta_{0}},\quad W=\frac{2}{k_{ 0}\sqrt{\Delta\epsilon\sin 2\theta_{0}}}, \tag{6}\]  
Figure 1: Sketch of nematic cell. A coherent light beam (yellow region) in which the electric field \(E\) is polarized in the \(Y\) direction propagates in the \(Z\) direction through a cell filled with a dye-doped nematic liquid crystal. Thin film electrodes (black) are deposited on the upper and lower cell walls (grey). An external low frequency voltage bias \(V_{LF}\) creates an electric field \(E_{LF}\) to pre-tilt the molecules at an angle \(\theta_{0}\) to \(Z\). The nematic molecules which are located at the boundaries are held tightly by the virtue of the anchoring films. The far right inset (black) dashed box exhibits the angular rotation of a nematic molecule with respect to the propagation direction \(Z\) in the absence (\(\theta_{0}\)) and presence (\(\theta_{0}+\phi\)) of the optical beam. and the resulting non-dimensional equations read  
\[i\frac{\partial u}{\partial z}+\frac{1}{2}\
abla^{2}u+2\phi u = 0, \tag{7}\] \[\
u\
abla^{2}\phi-2q\phi = -2|u|^{2}. \tag{8}\]  
Here, the dimensionless elasticity and pre-tilting parameters, \(\
u\) and \(q\), are given by  
\[2 ik_{0}n_{e}\frac{\partial E}{\partial Z}+\
abla^{2}E+k_{0}^{2}\Big{[}n_{ \perp}^{2}\cos^{2}\theta+n_{\parallel}^{2}\sin^{2}\theta-n_{\perp}^{2}\cos^{ 2}\theta_{0}-n_{\parallel}^{2}\sin^{2}\theta_{0}\Big{]}E=0, \tag{1}\]0  
Typical experimental beam parameter values are power \(P_{b}=2\) mW and half-width \(W_{b}=1.5\) um, with a wavelength \(\lambda_{0}=2\pi/k_{0}=1.064\) um in the near infrared [2; 5]. For the liquid crystal E7, a typical elastic constant is \(K=1.2\times 10^{-11}N\). These parameter values give the elasticity parameter \(\
u=O(100)\), as found in other studies [5; 40; 41]. This high value of \(\
u\) means that the nematic is operating in the highly nonlocal regime, in that the elastic response of the nematic extends far beyond the waist of the optical beam [2; 3; 4]. However, \(\
u\) is inversely proportional to the beam power \(P_{b}\). Note that, for \(\
u=0\), the nematic Equations (7) and (8) reduce to the standard NLS equation  
\[2 ik_{0}n_{e}\frac{\partial E}{\partial Z}+\
abla^{2}E+k_{0}^{2}\Big{[}n_{ \perp}^{2}\cos^{2}\theta+n_{\parallel}^{2}\sin^{2}\theta-n_{\perp}^{2}\cos^{ 2}\theta_{0}-n_{\parallel}^{2}\sin^{2}\theta_{0}\Big{]}E=0, \tag{1}\]1  
This is the local response limit for the nematic. Note that, in \((2+1)\)-dimensions, beams governed by this equation are unstable and can show catastrophic collapse above a critical power [17]. It is known that a nonlocal response, \(\
u\) large, stabilizes \((2+1)\)-dimensional optical beams [2; 3; 4; 5]. This is because the nematic response Equation (2) is elliptic, so its solution depends on \(u\) in the entire domain, the origin of the physical concept of nonlocality. Hence, by adjusting the beam power \(P_{b}\), the response of the medium can be adjusted from nonlocal to local, as long as the induced heating does not destroy the nematic phase at high power. The nematic system (7) and (8) is a focusing NLS-type system, that is the refractive index in the dimensional Equation (1) increases with beam intensity \(|u|^{2}\). Since focusing NLS equations do not possess (stable) DSW solutions, the equation needs to be defocusing; in such a case, the refractive index decreases with beam intensity, and DSW solutions do exist [19].