# Nematic Dispersive Shock Waves from Nonlocal to Local
## 2 Nematic Equations
### Derivation of the Extended KdV and Kawahara Equations  
We seek solutions of Equations (15)-(17) in the form of the following asymptotic expansions in the formal small parameter \(\varepsilon\equiv\sqrt{u_{i}-u_{+}}\) (with \(0<\varepsilon\ll 1\)):  
\[|u|^{2}=\rho=\rho_{+}+\varepsilon^{2}\rho_{1}(\xi,\eta)+\epsilon ^{4}\rho_{2}(\xi,\eta)+\cdots, \tag{26}\] \[v=\varepsilon^{2}V_{1}+\epsilon^{4}V_{2}+\epsilon^{6}V_{3}+\cdots,\] (27) \[\phi=\frac{\rho_{+}}{q}+\varepsilon^{2}\phi_{1}+\varepsilon^{4} \phi_{2}+\varepsilon^{6}\phi_{3}+\cdots, \tag{28}\]  
where \(\rho_{+}=u_{+}^{2}\), and the unknown functions \(\rho_{j}\), \(V_{j}\), and \(\phi_{j}\) (\(j=1,2,3,\ldots\)) depend on the stretched variables  
\[\xi=\varepsilon(x-Uz),\quad\eta=\varepsilon^{3}z. \tag{29}\]  
Here, \(U\) will be treated as an unknown velocity, which will be determined self-consistently. Substituting the expansions (26)-(28) into Equations (15)-(17), and using the stretched coordinates (29), we obtain a set of equations at the different orders in \(\varepsilon\). In particular, at the leading order, we derive the following linear equations:  
\[O(\varepsilon^{2}):\quad\rho_{1}-q\phi_{1}=0, \tag{30}\]  
we solve for \(\rho_{1}=P-\epsilon\rho_{2}\) and substitute the result into the above Equation (38). We, hence, obtain the nonlinear evolution equation for the field \(P(\xi,\eta)\)  
\[P_{\eta}+\frac{3}{2u_{+}}\sqrt{\frac{2}{q}}PP_{\xi}+\left(\frac{1 }{q}\right)^{3/2}\frac{4\
u u_{+}^{2}-q^{2}}{8\sqrt{2}u_{+}}P_{\xi\xi\xi}\] \[+\epsilon^{2}\Big{(}b_{1}P^{2}P_{\xi}+b_{2}P_{\xi}P_{\xi\xi}+b_{3 }PP_{\xi\xi\xi}+b_{4}P_{\xi\xi\xi\xi\xi\xi\xi}\Big{)}=0. \tag{40}\]  
The coefficients \(b_{j}\) (\(j=1,2,3,4\)) appearing in Equation (40) are given by  
\[b_{1} =-\frac{3}{8u_{+}^{3}}\sqrt{\frac{2}{q}},\quad b_{2}=-\left(\frac {1}{q}\right)^{3/2}\frac{20\
u u_{+}^{2}-13q^{2}}{32\sqrt{2}u_{+}^{3}},\] \[b_{3} =\left(\frac{1}{q}\right)^{3/2}\frac{4\
u u_{+}^{2}+q^{2}}{16 \sqrt{2}u_{+}^{3}},\quad b_{4}=\left(\frac{1}{q}\right)^{5/2}\frac{48\
u^{2} u_{+}^{4}+8\
u q^{2}u_{+}^{2}-q^{4}}{256\sqrt{2}u_{+}^{3}}.\]  
Then, we seek an asymptotic expansion in the optical beam intensity \(|u|\) as  
\[|u|=u_{+}+\epsilon^{2}Q+\cdots \tag{41}\]  
and use the relation \(|u|=\sqrt{\rho}\). This asymptotically gives \(P=2u_{+}Q\). The reductive nonlinear Equation (40) can now be written in terms of the field \(Q(\xi,\eta)\) as  
\[v=\varepsilon^{2}V_{1}+\epsilon^{4}V_{2}+\epsilon^{6}V_{3}+\cdots,\]0 \[v=\varepsilon^{2}V_{1}+\epsilon^{4}V_{2}+\epsilon^{6}V_{3}+\cdots,\]1  
The coefficients \(c_{j}\) (\(j=1,2,3,4\)) appearing in Equation (42) are given by  
\[v=\varepsilon^{2}V_{1}+\epsilon^{4}V_{2}+\epsilon^{6}V_{3}+\cdots,\]2  
Notice that Equation (42) is the so-called extended KdV equation (eKdV), which can model the evolution of steeper waves, with shorter wavelengths, than those governed by the KdV equation. As such, the eKdV equation has been used to describe solitary waves in plasmas [52] and shallow water waves [53] in the presence of higher order effects. We note that the coefficient of the third derivative dispersive term changes sign when  
\[v=\varepsilon^{2}V_{1}+\epsilon^{4}V_{2}+\epsilon^{6}V_{3}+\cdots,\]3  
Hence, in the high nonlocality, low power, limit, such that \(\
u>q^{2}/(4u_{+}^{2})\), the coefficient of the third derivative in eKdV equation is positive, so that its DSW (and solitary wave) solutions have positive polarity, with solitary waves at its leading edge and linear dispersive waves at its trailing edge. On the other hand, in the local limit, \(\
u<q^{2}/(4u_{+}^{2})\), the coefficient of the third derivative is negative and the DSW has negative polarity, with linear dispersive waves at its leading edge and solitary waves at its trailing edge, so that it resembles the standard NLS DSW [37]. The nematic DSW then undergoes a change of form from a KdV-type DSW to an NLS-type DSW as \(\
u\) decreases at the value of the nonlocality parameter given by (43). At this point, it is useful to make the following remarks.