# Nematic Dispersive Shock Waves from Nonlocal to Local
## 5 CDSW Regime  
For fixed a nonlocality parameter \(\
u\), as \(u_{+}\) decreases, the jump height \(u_{-}-u_{+}\) increases, the nematic DSW changes from RDSW to CDSW form; see Figure 3b. The reason for this is that, as the amplitude of the resonant wavetrain grows, it takes more conserved quantities from the DSW, so that its amplitude is reduced. In addition, the DSW becomes unstable in the CDSW regime, as for the Kahawara equation DSW [26], noting that, in the small jump height limit, the nematic equations reduce to the eKdV Equation (42), which becomes the Kahawara Equation (44) in the limit of large nonlocality \(\
u\). Figure 2c shows a typical nematic DSW in the CDSW regime. It can be seen that the DSW has changed from a modulated wavetrain with a monotonically decreasing amplitude from front to rear to a disordered wavetrain with an essentially uniform amplitude on average, except at its rear. This structure is in agreement with unstable DSW structure for the focusing NLS equation [56]. The unstable DSW can then be approximated by a train of equal amplitude solitary waves, which has been found to give good results for DSW solutions [57], particularly unstable DSWs, and for the particular case of the nematic CDSW in the high nonlocality, low optical power, regime [36]. The amplitude of the solitary waves of the CDSW is determined from mass and energy conservation equations for the underlying nonlinear dispersive wave equation [57]. If we set  
\[B_{2}=3\sqrt{\frac{2}{q}},\quad B_{3}=\frac{4\
u u_{+}^{2}-q^{2}}{8\sqrt{2}q^{3 /2}u_{+}} \tag{55}\]  
for simplicity, the eKdV Equation (42) has the mass conservation equation  
\[\frac{\partial}{\partial\eta}Q+\frac{\partial}{\partial\xi}\left[ \frac{1}{2}B_{2}Q^{2}+B_{3}Q_{\xi\xi}+\frac{1}{3}\varepsilon^{2}c_{1}Q^{3}+ \frac{1}{2}\varepsilon^{2}(c_{2}-c_{3})Q_{\xi}^{2}+\varepsilon^{2}c_{3}QQ_{ \xi\xi}\right.\] \[+\left.\varepsilon^{2}c_{4}Q_{\xi\xi\xi\xi\xi}\right]=0. \tag{56}\]  
Figure 5: Numerical solution of nematic Equations (11) and (12) in PDSW regime at \(z=1500\) for initial condition (13) with \(u_{+}=0.8\) and \(u_{-}=1.0\). Red (solid) line: \(|u|\); green (dashed) line: \(\phi\). Here, \(\
u=200\) and \(q=2\). The derivation of the energy conservation for the eKdV Equation (42) is not as straightforward. Multiplying the eKdV equation by \(Q\) and integrating by parts gives  
\[ \begin{split}&\tfrac{\partial}{\partial\eta}\tfrac{1}{2}Q^{2}+\tfrac {\partial}{\partial\xi}\Big{[}\tfrac{1}{8}B_{2}Q^{3}+B_{3}QQ_{gg}-\tfrac{1}{2 }B_{3}Q_{\xi}^{2}+\tfrac{1}{4}\varepsilon^{2}c_{1}Q^{4}+\tfrac{1}{2} \varepsilon^{2}c_{2}Q^{2}Q_{gg}^{2}\\ &\quad+\varepsilon^{2}c_{4}QQ_{gggg}-\varepsilon^{2}c_{4}Q_{\xi} Q_{gggg}+\tfrac{1}{2}\varepsilon^{2}c_{4}Q_{gg}^{2}\Big{]}+\varepsilon^{2} \Big{(}c_{3}-\tfrac{1}{2}c_{2}\Big{)}Q^{2}Q_{gggg}=0.\end{split}  \tag{57}\]  
The final term on the right hand side of this equation cannot be expressed as a perfect derivative. However, it can be approximately expressed in this form on noting that \(\varepsilon\) is small, so that, at first order, the eKdV Equation (42) is the KdV equation  
\[\frac{\partial Q}{\partial\eta}+B_{2}Q\frac{\partial Q}{\partial\xi}+B_{3} \frac{\partial^{3}Q}{\partial\xi^{3}}=0. \tag{58}\]  
We then have at leading order that  
\[\frac{\partial}{\partial\eta}Q^{3}=-3Q^{2}\big{(}B_{2}QQ_{\xi}+B_{3}Q_{gggg} \big{)}=-\frac{\partial}{\partial\xi}\frac{3}{4}B_{2}Q^{4}-3B_{3}Q^{2}Q_{gggg}. \tag{59}\]  
This expression may now be used to eliminate the term in \(QQ_{gggg}\) in Equation (57) to give the final energy conservation equation  
\[