# Nematic Dispersive Shock Waves from Nonlocal to Local
## 3 Nonlocal to Local Nematic DSWs  
The form and details of the DSW solution of the defocusing nematic Equations (11) and (12) will be found as the nonlocality \(\
u\) changes from \(O(100)\) to \(0\), that is from the nematic having a highly nonlocal response to a local response as the optical power increases. For high nonlocality, the DSW is of KdV type, with the leading edge of the DSW consisting of solitary waves of elevation [34; 35; 36]. The reason for this can be seen from the eKdV Equation (42) as for \(\
u>q^{2}/(4u_{+}^{2})\) the coefficient of the third derivative is positive, so that the DSW is of KdV type and has positive polarity. For \(0\leq\
u<q^{2}/(4u_{+}^{2})\), the sign of the third derivative is negative, and the DSW has negative polarity, with solitary waves at the trailing edge and linear dispersive waves at the leading edge, as for the NLS DSW [37]. A nematic DSW of NLS type is illustrated in Figure 2e for \(\
u=0.5\). Indeed, for \(\
u=0\), the nematic Equations (11) and (12) reduce to the standard NLS equation on substituting for \(\phi\) from (12) into (11). The DSW solution of the NLS equation is well known [37] as the NLS equation is integrable and so the solution is completely determined. Hence, the solution for this regime will not be considered here.  
The existence regions for the various nematic DSW types as the nonlocality parameter \(\
u\) varies (optical power varies), found from full numerical solutions of the nematic Equations (11) and (12) are shown in Figure 3 as the initial level ahead \(u_{+}\) varies. The range \(0.3\leq u_{+}\leq 0.9\) was chosen as this encompassed all six of the DSW types studied here. In addition, most of the theoretical expressions for the boundaries between these regions and the solutions within each region were based on \(u_{-}-u_{+}\) small, for example, the boundary (43), which is based on the eKdV Equation (42). It can be seen that, over most of the \((u_{+},\
u)\) domain, the nematic DSW is of CDSW or TDSW type, so that it is typically unstable. In addition, it is deduced that the nematic DSW is of NLS-type only for small values of the nonlocality parameter \(\
u\) of \(2\) and below. The nematic DSW is then nonlocal, except for high enough optical powers for which \(\
u\) is small. The nonlocality parameter \(\
u\) is given by (9). For the nematic liquid crystal 4-(trans-4-n-hexylcyclohexyl)-isothiocyanato-benzene (6 CHBT), the parameter values are \(K\sim 10^{-11}N\), \(n_{\parallel}=1.6335\) and \(n_{\perp}=1.4967\)[54]. Let us take the pre-tilt angle \(\theta_{0}\) to be \(\pi/4\) so that the nematic response is maximized [2]. A typical beam wavelength is 1064 nm, and a typical half width \(W_{b}\) is 1.5 \(\upmu\)m [54]. With these parameter values, it is found that \(\
u=2\) when the beam power \(P_{b}\) is 288 mW, far in excess of typical beam powers of a few milliWatts to a few tens of milliWatts [2; 54]. Such a large optical power can result in the nematic medium being heated enough so that its temperature goes above the critical temperature, 43 \({}^{\circ}\)C for 6 CHBT [54], so that it undergoes a phase change out of the nematic state. In this regard, it should be noted that experimental nematic cells are small, of the order of 1 mm in the down cell propagation direction of the beam and 100 \(\upmu\)m \(\times\) 10 mm in cross-section. The thin cross-section is the direction in which the pre-tilting electric field is applied, which results in a stable and uniform molecular pre-tilt. We then deduce that, for experimental beam powers, the nematic bore will be in the nonlocal response regime with \(\
u\) large, which is low optical power.