# Nematic Dispersive Shock Waves from Nonlocal to Local
## 6 TDSW Regime
The Lagrangian for the hydrodynamic form of the nematic Equations (15)-(17) is  
\[L=-2\rho\psi_{z}-\frac{1}{4}\frac{\rho_{x}^{2}}{\rho}-\rho\psi_{x}^{2}-4\rho \phi+\
u\phi_{x}^{2}+2q\vartheta^{2}. \tag{82}\]  
Applying Nother's Theorem we have that translation invariance with respect to the phase \(\psi\) gives the mass conservation law  
\[\frac{\partial}{\partial z}\frac{\partial L}{\partial\psi_{z}}+\frac{\partial }{\partial x}\frac{\partial L}{\partial\psi_{x}}=0, \tag{83}\]  
translation invariance with respect to space \(x\) yields the momentum conservation law  
\[v = \bar{v}+a\
u_{1}\cos\varphi+\ldots,\]0  
and translation invariance with respect to time-like \(z\) gives the energy conservation law  
\[v = \bar{v}+a\
u_{1}\cos\varphi+\ldots,\]1  
Substituting the Stokes expansions (74)-(76) into these conservation laws and averaging by integrating in \(\varphi\) from \(0\) to \(2\pi\)[12] yields the modulation equations, truncated to \(O(1/\
u)\),  
\[v = \bar{v}+a\
u_{1}\cos\varphi+\ldots,\]2 \[v = \bar{v}+a\
u_{1}\cos\varphi+\ldots,\]3 (87) \[v = \bar{v}+a\
u_{1}\cos\varphi+\ldots,\]4  
\[ \begin{array}{l}\frac{\partial}{\partial z}\bigg{(}\tilde{\rho}\tilde{v}^{2} +\frac{2\tilde{\rho}^{2}}{q}+\frac{k^{2}a^{2}}{4\tilde{\rho}}+\frac{k^{2}a^{2} }{2\tilde{\rho}}+\frac{4a^{2}}{vk^{2}}+\frac{4a^{2}}{vk^{3}}\bigg{)}+\frac{ \partial}{\partial x}\bigg{(}\tilde{\rho}\tilde{v}^{3}+\frac{4\tilde{\rho}^{2} }{q}+\frac{k^{2}a^{2}}{q}+\frac{k^{3}a^{2}}{4\tilde{\rho}}\\ +\frac{3\tilde{o}^{2}k^{2}}{4\tilde{\rho}}+\frac{3\tilde{o}^{2}k^{2}}{4\tilde{ \rho}}+\frac{6\tilde{o}^{2}}{vk^{2}}+\frac{\tilde{o}^{2}a^{2}}{vk^{3}}+\frac {2a^{2}}{vk}\bigg{)}=0\end{array}  \tag{89}\]  
for the (slowly varying) amplitude \(a\), wavenumber \(k\), and means \(\tilde{\rho}\) and \(\tilde{v}\) of the modulated Stokes wave [12]. The modulation Equation (87) is that, for optical power conservation, Equation (88) is momentum conservation, and (89) is energy conservation. The modulation Equation (86)is the equation for conservation of waves, \(k_{z}+\omega_{x}=0\), on noting that the \(x\) derivative term is just the dispersion relation (77). In this regard, it should be noted that the dispersion relation for the Stokes' wave from which the modulation equations are calculated has the carrier waves' phase shift term \(2\bar{\rho}/q\) added [35; 36], as explained above,  
\[\omega=k\bar{v}+\frac{1}{2}k^{2}+\frac{2\bar{\rho}}{q}+\frac{4\bar{\rho}}{vk^{2 }}-\frac{k^{2}a^{2}}{8\beta^{2}}-\frac{3a^{2}}{vk^{2}\bar{\rho}}. \tag{90}\]  
Figure 2d shows that, in the TDSW regime, the KdV-type nematic bore structure of Figure 2a-c has disappeared, leaving a dominant resonant wavetrain which is linked to the intermediate level \(u_{i}\) by a negative polarity solitary wave [26]. As discussed above, this link between the resonant wavetrain and the intermediate level can be approximated by a Whitham shock, a jump in the modulation equation variables [36; 48], so that the Whitham shock links the resonant wavetrain with the level \(u_{i}\) behind, in a similar manner to a gas dynamic shock wave links two compressible flow states [12]. Let us denote the amplitude, wavenumber, mean level and mean phase gradient of the resonant wavetrain by \(a_{r}\), \(k_{r}\), \(\bar{\rho}_{r}\), and \(\bar{\rho}_{r}\), respectively. Matching the Whitham shock velocity \(U_{s}\) to the Stokes' wave velocity (90), as these are co-propagating, gives  
\[U_{s}=\bar{\sigma}_{r}+\frac{1}{2}k_{r}+\frac{2\bar{\rho}_{r}}{qk_{r}}+\frac{4 \bar{\rho}_{r}}{vk_{r}^{2}}-\frac{k_{r}a_{r}^{2}}{8\bar{\rho}_{r}^{2}}-\frac{ 3a_{r}^{2}}{vk_{r}^{3}\bar{\rho}_{r}}. \tag{91}\]  
Ahead of the Whitham shock, there is the resonant wavetrain, and behind it is a flat shelf, the intermediate level, which is a wavetrain of zero amplitude. The mass, momentum, and energy conservation Equations (87)-(89) then give the jump conditions  
\[