# Nematic Dispersive Shock Waves from Nonlocal to Local
## 6 TDSW Regime
\begin{array}{l}U_{s}(\bar{\rho}_{r}-\rho_{i})=\bar{\rho}_{r}\bar{\rho}_{r}+ \frac{k_{r}a_{r}^{2}}{4\bar{\rho}_{r}}+\frac{2a_{r}^{2}}{vk_{r}^{2}}-\rho_{i} \bar{\
u}_{i},\\ U_{s}\Big{(}\bar{\rho}_{r}\bar{\rho}_{r}+\frac{k_{r}a_{r}^{2}}{4\bar{\rho}_{r} }+\frac{2a_{r}^{2}}{vk_{r}^{2}}-\rho_{i}\bar{\
u}_{i}\Big{)}=\frac{\bar{\rho} _{r}^{2}}{q}+\bar{\rho}_{r}\bar{\rho}_{r}^{2}+\frac{k_{r}^{2}a_{r}^{2}}{4\bar{ \rho}_{r}}+\frac{\bar{\rho}_{r}k_{r}a_{r}^{2}}{4\bar{\rho}_{r}}+\frac{4\bar{ \rho}_{r}a_{r}^{2}}{vk_{r}^{2}}\\ +\frac{a_{r}^{2}}{vk_{r}^{2}}-\frac{\bar{\rho}_{r}^{2}}{q}-\rho_{i}\bar{\
u}_{ i}^{2},\end{array}  \tag{92}\]  
\[ \begin{array}{l}U_{s}\Big{(}\bar{\rho}_{r}\bar{\rho}_{r}^{2}+\frac{2\bar{ \rho}_{r}^{2}}{q}+\frac{k_{r}^{2}a_{r}^{2}}{4\bar{\rho}_{r}}+\frac{k_{r}\rho_{r }a_{r}^{2}}{2\bar{\rho}_{r}}+\frac{4a_{r}^{2}}{vk_{r}^{2}}+\frac{4\bar{\rho}_{ r}a_{r}^{2}}{4\bar{\rho}_{r}^{2}}-\rho_{i}\bar{\
u}_{i}^{2}-\frac{2\bar{\rho}_{ r}^{2}}{q}\Big{)}=\bar{\rho}_{r}\bar{\rho}_{r}^{3}+\frac{4\bar{\rho}_{r}\bar{\rho}^{2}}{q}\\ +\frac{k_{r}a_{r}^{2}}{q}+\frac{k_{r}^{3}a_{r}^{2}}{4\bar{\rho}_{r}}+\frac{3 \bar{\rho}_{r}k_{r}^{2}a_{r}^{2}}{4\bar{\rho}_{r}}+\frac{3\bar{\rho}_{r}k_{r}a _{r}^{2}}{4\bar{\rho}_{r}}+\frac{6a_{r}^{2}a_{r}^{2}}{vk_{r}^{2}}+\frac{2a_{r} ^{2}}{vk_{r}^{2}}-\rho_{i}\bar{\
u}_{i}^{3}-\frac{4\bar{\
u}_{i}\bar{\rho}_{r} ^{2}}{q}\end{array}  \tag{93}\]  
across the Whitham shock. Together with the resonance condition (91), these form four equations for the five unknowns \(U_{s}\), \(a_{r}\), \(k_{r}\), \(\bar{\rho}_{r}\), and \(\bar{\
u}_{r}\), noting that \(\rho_{i}\) and \(\bar{\
u}_{i}\) are given by (24) and (25), respectively. The final equation is obtained by assuming that the Riemann invariant \(R_{-}\) (22) is conserved through the resonant wavetrain and its lead partial DSW, which is valid for a full DSW [19; 50]. This then determines the mean of the resonant phase gradient \(\bar{\sigma}_{r}\)[36]. This Riemann invariant condition gives  
\[\bar{\sigma}_{r}=2\sqrt{\frac{2}{q}}\big{(}\sqrt{\bar{\rho}_{r}}-\sqrt{\bar{ \rho}_{+}}\big{)}=2\sqrt{\frac{2}{q}}\big{(}\sqrt{\bar{\rho}_{r}}-u_{+}\big{)}. \tag{94}\]  
The above nematic Whitham modulation equation jump conditions (92)-(93) with (94) can be solved numerically for \(U_{s}\), \(a_{r}\), \(k_{r}\), and \(\bar{\rho}_{r}\) using [PERSON]'s method. The full details for this numerical solution of the Whitham shock jump conditions can be found in Reference [36]. Figure 8 shows comparisons for the Whitham shock velocity from the nonlocal to local limits, the optical power increasing, as given by the jump conditions and full numerical solutions of the nematic Equations (11) and (12). The values of the level ahead \(u_{+}\) were chosen to lie in the TDSW regime; see Figure 3. It can be seen that there is excellent agreement between the theoretical and numerical solutions from high nonlocality, \(\
u\) large, down to \(\
u=O(10)\). As for the lead solitary wave amplitude comparison of Figure 4,there is little change in the Whitham shock velocity as the nonlocality parameter \(\
u\) varies by an order of magnitude, with only a small increase in the velocity. There is a small, increasing deviation between the theoretical and numerical values towards \(\
u=20\). This is due to the onset on the VDSW regime for which \(u\) vanishes at a point, a vacuum point [35; 36]. Once the vacuum point is reached, \(|u|\) cannot decrease further, so that the Whitham shock jump conditions need to be modified [36]. This will not be pursued further here. For a fixed level ahead \(u_{+}\), as the nonlocality parameter \(\
u\) decreases (optical power increases), the nematic DSW evolves from CDSW to TDSW type; see Figure 3. The borderline between the CDSW and TDSW regimes can be determined from the Whitham shock jump conditions (92)-(93) and the resonance condition (91) based on the following condition.