# Nematic Dispersive Shock Waves from Nonlocal to Local
## 4 PDSW and RDSW Regimes
It can then be found from the characteristics (48) that the DSW lies in the range  
\[s_{i}=\sqrt{\frac{2}{q}}\bigg{\{}4u_{+}-3u_{i}-(u_{i}-u_{+})^{2 }\bigg{(}\frac{1}{2}C_{1}+C_{3}-\frac{64}{3}C_{4}\bigg{)}\bigg{\}}\\ \leq\frac{x}{z}\leq\sqrt{\frac{2}{q}}\bigg{\{}2u_{i}-u_{+}+\frac{1 }{3}(u_{i}-u_{+})^{2}(C_{1}-C_{3}+4C_{4})\bigg{\}}=s_{+}, \tag{50}\]  
where \(s_{i}\) and \(s_{+}\) are the harmonic and solitary wave edge velocities of the DSW, respectively. Comparisons between the lead solitary wave amplitude \(a_{+}\) as given by the asymptotic DSW solution (45), with \(m=1\), and numerical solutions are given in Figure 4 as the nonlocality parameter \(\
u\) varies. The existence regions for the PDSW and RDSW types depend on both \(u_{+}\) and \(\
u\), as shown in Figure 3, so that the comparison curves for each \(u_{+}\) were stopped at the boundary between the RDSW and CDSW regimes. Figure 4 shows the lead solitary wave amplitude for the full eKdV Equation (42) and the Kawahara equation,which is (42) with \(c_{1}=c_{2}=c_{3}=0\). Previous work on the nematic bore [35; 36] was based on the Kawahara equation, that is only the higher order fifth derivative was included in the asymptotic eKdV equation, so the lead wave amplitude based on this equation is given in the figure to determine the effect of the extra higher order terms in the full eKdV Equation (42). A key observation is that the height of the lead wave of the DSW depends very weakly on the strength of the nonlocality, with little variation even down to \(v=O(10)\) from the high nonlocality amplitudes with \(\
u=O(100)\). In some sense, the nematic DSW is then nonlocal down to small values of \(\
u\) in the PDSW and RDSW regimes, which was also deduced above from Figure 3. It can also be seen that the inclusion of the extra higher order terms in the eKdV Equation (42) over the Kawahara equation improves the agreement with numerical solutions on the whole, especially as the nonlocality parameter \(\
u\) decreases, but the effect of these extra terms is small, with the Kawahara equation giving good agreement over the whole range of \(\
u\) and for all values of \(u_{+}\), except near the RDSW/CDSW borderline at \(u_{+}=0.7\). This is expected as the weak dependence on the nonlocality parameter \(\
u\) means that the DSW is nonlocal, so that \(\
u\) can be taken as large. The dominant higher order term in the eKdV equation is \(\varepsilon^{2}c_{4}Q_{\xi\xi\xi\xi\xi}\) in this limit, as noted in (44). As \(u_{+}=0.7\) is approached the resonant radiation shed by the DSW is of relatively large amplitude as in this limit the RDSW/CDSW boundary is approached. This results in oscillations in the lead wave amplitude as the resonant radiation moves through the DSW and is shed. In these cases, the numerical amplitude shown in Figure 4 was calculated as an average in \(z\) over the last few amplitude oscillations in the numerical solution. It is noted that except for \(u_{+}=0.75\) and \(0.7\) the amplitude grows as \(\
u\) decreases. This change in behavior is due to the DSW changing form as it transitions from the RDSW to the CDSW regime, for which the resonant radiation has a major effect on the DSW with its amplitude decreasing markedly due to the large amount of mass being shed as resonant radiation [36]. Figure 4: Comparison between lead wave amplitude \(a_{+}\) of PDSW and RSDW with numerical solutions of the nematic Equations (11) and (12) with the initial condition (13). Numerical amplitude: red squares; amplitude (45) with \(m=1\) given by undular bore solution of eKdV Equation (42): red (solid) line; amplitude given by undular bore solution of Kawahara equation (Equation (42) with \(c_{1}=c_{2}=c_{3}=0\)): green (dashed) line. Here, \(u_{-}=1.0\) and \(q=2\). The full linear dispersion relation (18) for the nematic Equations (11) and (12) is  
\[\omega=k\vartheta+\frac{\sqrt{\varrho}k}{\sqrt{vk^{2}+2q}}\bigg{[}\frac{vk^{2}+2 q}{4\varrho}k^{2}+4\bigg{]}^{1/2}+\frac{2\bar{\varrho}}{q} \tag{51}\]  
when the mean \(2\bar{\varrho}/q\) is added.