# Nematic Dispersive Shock Waves from Nonlocal to Local
## 5 CDSW Regime
\begin{split}&\tfrac{\partial}{\partial\eta}\Big{[}\tfrac{1}{2}Q^{2}- \varepsilon^{2}\tfrac{c_{3}-\tfrac{1}{4}c_{2}}{3B_{3}}Q^{3}\Big{]}+\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}\\ &\quad+\tfrac{1}{4}\varepsilon^{2}c_{2}Q^{2}Q_{gg}+\varepsilon^{2 }c_{4}Q_{QQ_{gggg}}-\varepsilon^{2}c_{4}Q_{\xi}Q_{gggg}+\tfrac{1}{2}\varepsilon ^{2}c_{4}Q_{\xi}^{2}\\ &\quad-\varepsilon^{2}\tfrac{B_{2}}{4B_{3}}\big{(}c_{3}-\tfrac{1} {2}c_{2}\Big{)}Q^{4}\Big{]}=0,\end{split}  \tag{60}\]  
which is accurate to \(O(\varepsilon^{2})\). To obtain an approximation to the nematic CDSW, let us assume that, at position \(\eta\), the CDSW consists of \(N\) equal solitary waves of amplitude \(\tilde{a}_{s}\) and width \(\tilde{w}_{s}\)[57], where we shall use tildes to denote scaled variables in the moving and stretched coordinates \((\tilde{z},\eta)\). It is also assumed that the CDSW sheds a uniform downstream resonant wavetrain of (scaled) amplitude \(\tilde{a}_{r}\). Then, as \(\xi\to-\infty\), \(Q\to 1\) and as \(\xi\to\infty\), \(Q\to\tilde{a}_{r}\cos\big{(}\tilde{k}_{r}\xi-\tilde{\omega}_{r}\eta\big{)}\), since \(|u|=u_{+}+\varepsilon^{2}Q\) with \(\varepsilon^{2}=u_{i}-u_{+}\). As the CDSW is approximated by a train of solitary waves, the solitary wave solution of the eKdV Equation (42) is also needed. While there is no known exact solitary wave solution of this equation, there is an asymptotic solution for \(\varepsilon\ll 1\)[58]. To use this solution, the eKdV Equation (42) needs to be rescaled to conform with the eKdV scaling of Reference [58]. Performing this, we find that the asymptotic solitary wave solution of the eKdV Equation (42) is  
\[Q=\gamma_{1}\operatorname{sech}^{2}\frac{\xi-V_{s}\eta}{W}+\gamma_{2} \operatorname{sech}^{4}\frac{\xi-V_{s}\eta}{W},\quad V_{s}=\frac{1}{3}B_{2}A \Big{(}1+2\varepsilon^{2}C_{4}A\Big{)}, \tag{61}\]  
where  
\[W=\frac{\sqrt{12B_{3}}}{\sqrt{B_{2}A}},\quad\gamma_{1}=A+\varepsilon^{2}C_{6} A^{2},\quad\gamma_{2}=\varepsilon^{2}C_{7}A^{2}. \tag{62}\]  
The rescaled coefficients \(c_{i}\), \(i=1,\dots,4\), of the eKdV Equation (42), denoted by \(C_{i}\), \(i=1,\dots,4\), are given by  
\[C_{1}=\frac{6}{B_{2}}c_{1},\quad C_{2}=\frac{1}{B_{3}}c_{2}, \quad C_{3}=\frac{1}{B_{3}}c_{3},\quad C_{4}=\frac{B_{2}}{6B_{3}^{2}}c_{4},\] \[C_{6}=\frac{2}{3}C_{3}-\frac{1}{6}C_{1}+\frac{1}{6}C_{2}-5C_{4}, \quad C_{7}=\frac{15}{2}C_{4}-\frac{1}{2}C_{3}+\frac{1}{12}C_{1}-\frac{1}{4}C _{2}. \tag{63}\]  
It is noted that these scaled \(C_{i}\), \(i=1,\dots,4\), are the same as the \(C_{i}\) (49) used for the perturbed DSW solution (45)-(48) due to the same rescaling from the eKdV Equation (42) being used for this solution. The mass and energy conservation Equations (56) and (60) can now be used to determine the amplitude of the solitary waves in the nematic CDSW in the equal amplitude approximation. The mass and energy conservation equations are integrated in \(\xi\) from \(-\infty\) to \(\infty\). The integral of the mass and energy densities are approximated by \(N\) times these for a single solitary wave (61). The flux terms are calculated using the boundary conditions stated above, \(Q=1\) at \(\xi=-\infty\) and \(Q=\bar{u}_{r}\cos(\bar{k}_{r}\xi-\bar{\omega}_{r}\eta)\) at \(\xi=\infty\). The resonant radiation flux at \(\xi=\infty\) is calculated by averaging the periodic radiation over a period [36].