# Nematic Dispersive Shock Waves from Nonlocal to Local
## 2 Nematic Equations
The nematic medium can feature a defocusing response through the addition of azo-dyes [33]. The change in the nematic response due to the addition of the azo-dye is physically complicated, with the \"order parameter\" change being opposite to that in the absence of the dye. A simple model of this response change is to modify the electric field Equation (7) from focusing to defocusing. In addition, the analysis of DSWs is simplest in \((1+1)\)-dimensions as then there are no geometric spreading effects. With these assumptions and simplifications, the nematic equations become  
\[2 ik_{0}n_{e}\frac{\partial E}{\partial Z}+\
abla^{2}E+k_{0}^{2}\Big{[}n_{ \perp}^{2}\cos^{2}\theta+n_{\parallel}^{2}\sin^{2}\theta-n_{\perp}^{2}\cos^{ 2}\theta_{0}-n_{\parallel}^{2}\sin^{2}\theta_{0}\Big{]}E=0, \tag{1}\]2 \[2 ik_{0}n_{e}\frac{\partial E}{\partial Z}+\
abla^{2}E+k_{0}^{2}\Big{[}n_{ \perp}^{2}\cos^{2}\theta+n_{\parallel}^{2}\sin^{2}\theta-n_{\perp}^{2}\cos^{ 2}\theta_{0}-n_{\parallel}^{2}\sin^{2}\theta_{0}\Big{]}E=0, \tag{1}\]3  
The same system of equations also describes optical beam propagation in thermal optical media for which the refractive index depends on the temperature of the medium [42]. Such thermal optical media typically have a defocusing response. For these defocusing nematic equations, a suitable initial condition which will generate a DSW is the intensity jump initial condition \[u=\left\{ \begin{array}{ll}u_{-,}&x<0\\ u_{+,}&x>0\end{array} \right.,\qquad\phi=\left\{ \begin{array}{ll}\frac{u^{2}}{q}, &x<0\\ \frac{u^{2}}{q},&x>0\end{array} \right.. \tag{13}\]  
The DSW solution of the defocusing nematic Equations (11) and (12) has been studied in the highly nonlocal limit \(v\gg 1\) in previous work [34; 35; 36]. As stated above, in the local limit \(v\ll 1\), this system reduces to a perturbed defocusing NLS equation and becomes the defocusing NLS equation for \(v=0\). The DSW solution of the defocusing NLS equation is known [37], so that the perturbed local DSW solution can be found using perturbed Whitham modulation theory [43]. Before studying the behavior of the nematic DSW as \(v\) varies from large to small, some previously derived results [34; 35; 36] will be briefly summarized. The analytical DSW solutions derived in this work will be compared with full numerical solutions of the nematic Equations (11) and (12) with the initial condition (13). The electric field Equation (11) was solved using the pseudo-spectral method of Fornberg and Whitham [24], as extended [44; 45] to improve the stability for high wavenumbers through the use of an integrating factor. The \(x\) derivatives were calculated using the Fast Fourier Transform (FFT), and the solution was advanced in \(z\) in Fourier space employing the fourth-order Runge-Kutta method, as detailed in previous work [36; 46]. The step initial condition (13) was smoothed using the hyperbolic tangent function, as detailed in Reference [36; 46]. The director Equation (12) was also solved using the FFT, as the equation does not have a singularity at zero wavenumber due to the \(2q\theta\) term. Figure 2 displays a summary of the nematic DSW types as the nonlocality \(v\) varies from large \(v\), corresponding to a highly nonlocal response for low beam power, to small \(v\), pertinent to a local medium response for high beam power. The terminology for the DSW regimes will be taken from previous work on radiating DSWs [26; 36]. In particular, the DSW regimes from nonlocal to local response will now be detailed, for \(u_{-}=1\) and \(u_{+}=0.8\), with the existence intervals for the various DSW types given in Table 1.  
* PDSW (perturbed DSW): This regime is illustrated in Figure 2a. The nematic DSW is essentially a KdV DSW governed by the Kawahara Equation (44) and its solution can be found as a perturbed KdV DSW [36; 47]. This DSW regime is non-resonant and there is no resonant wavetrain attached to the leading edge of the DSW.