# Nematic Dispersive Shock Waves from Nonlocal to Local
## 2 Nematic Equations
As noted, the first four DSW regimes also occur for the Kahawara equation and the nematic equations in the low power, high nonlocality limit. The standard method to analyze DSW solutions of nonlinear dispersive wave equations of defocusing NLS-type is to transform the equation into hydrodynamic form using the Madelung transformation [19]  
\[u=\sqrt{\rho}e^{i\psi},\quad v=\psi_{x}, \tag{14}\]  
where the real functions \(\rho\) and \(\psi\) denote the density and phase of the field \(u\), while \(v\) is the fluid velocity. Upon substituting, it is found that the nematic Equations (11) and (12) become  
\[\frac{\partial\rho}{\partial z}+\frac{\partial}{\partial x}(\rho v) = 0, \tag{15}\] \[\frac{\partial v}{\partial z}+v\frac{\partial v}{\partial x}+2 \frac{\partial\phi}{\partial x}-\frac{\partial}{\partial x}\left(\frac{\rho_ {xx}}{4\rho}-\frac{\rho_{x}^{2}}{8\rho^{2}}\right) = 0,\] (16) \[v\frac{\partial^{2}\phi}{\partial x^{2}}-2q\phi = -2\rho. \tag{17}\]  
The nematic equations are characterized by the linear dispersion relation [34; 35]  
\[\omega=k\bar{\sigma}+\frac{\sqrt{\bar{\rho}\bar{k}}}{\sqrt{\
u k^{2}+2q}}\left[ \frac{vk^{2}+2q}{4\bar{\rho}}k^{3}+4k\right]^{1/2}, \tag{18}\]  
for waves around the mean level \(\bar{\rho}\) for \(\rho\) and \(\bar{v}\) for \(v\). In the short wave and high nonlocality limit, \(vk^{2}\gg 1\), this dispersion relation can be approximated by  
\[\omega=k\bar{\sigma}+\frac{1}{2}k^{2}+\cdots. \tag{19}\]  
In the opposite local limit with \(\
u\) small, the dispersion relation can be expanded as  
\[\omega=k\bar{\sigma}+k^{2}\left[\frac{1}{4}k^{2}+\frac{2}{q}\bar{\rho}\right] ^{1/2}-\frac{v\bar{\rho}k^{3}}{2q^{2}}\left[\frac{1}{4}k^{2}+\frac{2}{q}\bar{ \rho}\right]^{-1/2}+\cdots. \tag{20}\]  
As expected, at the leading-order, \(O(1)\), this dispersion relation is the same as that for the NLS equation [19; 50]. These dispersion relations are needed for the determination of the resonant wavetrain generated by the nematic DSW by which the linear phase velocity is matched to the velocity of the DSW front. As seen from Figure 2, the solution outside of the DSW and the resonant wavetrain is non-dispersive. On neglecting dispersion, the nematic Equations (15)-(17) become the \begin{table}
\begin{tabular}{c c} \hline PDSW & \(88<\
u\) \\ RDSW & \(34<\
u<88\) \\ CDSW & \(4.1<\
u<34\) \\ TDSW & \(1.53<\
u<4.1\) \\ RNLS DSW & \(0.60<\
u<1.53\) \\ NLS DSW & \(0\leq\
u<0.60\) \\ \hline \end{tabular}
\end{table} 
Table 1: Regime boundaries for type classifications of Figure 2. Here, \(u_{-}=1.0\), \(u_{+}=0.8\) and \(q=2\). shallow water equations [12], with \(\rho\) playing the role of fluid depth and \(v\) the (horizontal) fluid velocity. In Riemann invariant form, the dispersionless nematic equations read:  
\[v+\frac{2\sqrt{2}}{\sqrt{q}}\sqrt{\rho}=R_{+}=\text{constant}\quad\text{on} \quad C_{+}:\frac{dx}{dz}=V_{+}=v+\frac{\sqrt{2}}{\sqrt{q}}\sqrt{\rho}, \tag{21}\]  
\[v-\frac{2\sqrt{2}}{\sqrt{q}}\sqrt{\rho}=R_{-}=\text{constant}\quad\text{on} \quad C_{-}:\frac{dx}{dz}=V_{-}=v-\frac{\sqrt{2}}{\sqrt{q}}\sqrt{\rho}. \tag{22}\]  
The initial level behind \(u_{-}\) is linked to the intermediate shelf by a simple wave on the characteristic \(C_{-}\). This simple wave solution has been derived previously [34; 35; 36] and is of the form:  
\[|u|=\sqrt{\rho}=\left\{ \begin{array}{cc}u_{-},&\frac{x}{z}<-\frac{\sqrt{2}u_ {-}}{\sqrt{q}}\\ \frac{\sqrt{q}}{3\sqrt{2}}\left[\frac{2\sqrt{2}u_{-}}{\sqrt{q}}-\frac{x}{z} \right],&-\frac{\sqrt{2}u_{-}}{\sqrt{q}}\leq\frac{x}{z}\leq\frac{\sqrt{2}}{ \sqrt{q}}\big{(}2u_{-}-3\sqrt{\rho_{i}}\big{)},\\ \sqrt{\rho_{i}},&\frac{\sqrt{2}}{\sqrt{q}}\big{(}2u_{-}-3\sqrt{\rho_{i}}\big{)} <\frac{x}{z}\leq s_{i}\end{array} \right. \tag{23}\]  
with \(v=2\sqrt{2}(u_{-}-\sqrt{\rho})/\sqrt{q}\), where \(s_{i}\) is the velocity of the trailing edge of the DSW which lies on the intermediate level \(u_{i}\).