# Nematic Dispersive Shock Waves from Nonlocal to Local
## 6 TDSW Regime  
Figure 2d displays a typical DSW in the TDSW regime. There is (almost) a constant amplitude resonant wavetrain which, at its trailing edge, is connected to the intermediate level \(u_{i}\). At its leading edge there is a modulated wavetrain which takes \(u\) down to the level \(u_{+}\) ahead. This wavetrain leading the resonant wave is a partial DSW [27; 36; 49]. A partial DSW differs from a standard DSW in that it connects a uniform state to a periodic wavetrain, unlike a standard DSW which links two different levels. While there is a negative polarity solitary wave connecting the resonant wavetrain to the intermediate level [26], this connection can be approximated by a Whitham shock [36; 48], a shock, a jump, in the modulation parameters of the wavetrain, wavelength, frequency, amplitude, and mean level, of the Whitham modulation equations for the modulated periodic wavetrain [12]. A Whitham shock is determined from the Whitham modulation equations. As noted above, the nematic DSW is in the nonlocal regime, so the appropriate Whitham modulation equations are those for \(\
u\) large. u\gg 1\) (low optical power) have previously been determined [36], so these modulation equations will just be quoted here. These modulation equations determine the mean level \(\bar{\rho}\) of \(\rho\), the amplitude \(a\) and the wavenumber \(k\) of the Stokes' wave solution of the nematic Equations (15)-(17). As there is no known general periodic wave solution of the nematic equations, the highly nonlocal Whitham modulation equations are derived based on the weakly nonlinear Stokes wavetrain for the nematic equations [36]. It can be seen from Figure 2d that the resonant wavetrain has small amplitude, so the weakly nonlinear limit is appropriate. In the highly nonlocal limit \(\
u\gg 1\), the Stokes' wave solution of the nematic Equations (15)-(17) is  
\[\rho = \bar{\rho}+a\cos\varphi+\ldots, \tag{74}\] \[v = \bar{v}+a\
u_{1}\cos\varphi+\ldots,\] (75) \[\phi = \frac{\bar{\rho}}{q}+a\phi_{1}\cos\varphi+\ldots,\] (76) \[\omega = \omega_{0}+a\omega_{1}+a^{2}\omega_{2}+\ldots, \tag{77}\]  
where the uniform phase is \(\varphi=kx-\omega z\), and the over bar, \(\bar{\rho}\) and \(\bar{v}\), denotes the mean value of a wave parameter [36]. The amplitude \(a\) of the (Stokes) wave is assumed to be small. The work of Reference [36] gives that, at \(O(a)\), the nematic Stokes coefficients and the \(O(a^{2})\) correction to the dispersion relation are  
\[\omega_{0} = k\vartheta+\frac{k^{2}}{2}+\frac{4\vartheta}{vk^{2}}-\frac{8 \beta q}{v^{2}k^{4}}-\frac{16\vartheta^{2}}{v^{2}k^{6}}+\ldots, \tag{78}\] \[v_{1} = \frac{k}{2\vartheta}+\frac{4}{vk^{3}}-\frac{8q}{v^{2}k^{5}}-\frac {16\vartheta}{v^{2}k^{7}}+\ldots,\] (79) \[\phi_{1} = \frac{2}{vk^{2}}-\frac{4q}{v^{2}k^{4}}+\ldots,\] (80) \[\omega_{2} = -\frac{k^{2}}{8\vartheta^{2}}-\frac{3}{\rho vk^{2}}+\frac{6q}{ \vartheta v^{2}k^{4}}-\frac{20}{v^{2}k^{6}}+\ldots \tag{81}\]  
The coefficient \(\omega_{1}\) is set to zero, \(\omega_{1}=0\), to eliminate secular terms. The weakly nonlinear Whitham modulation equations can then be derived by averaging conservation laws deduced from Nother's Theorem [59].