# Nematic Dispersive Shock Waves from Nonlocal to Local
## 4 PDSW and RDSW Regimes  
Typical PDSW and RDSW solutions are illustrated in Figure 2a,b. In the PDSW regime, the DSW is not in resonance with linear diffractive radiation, so that the DSW is of KdV type, as seen in the boundary (43), and is a perturbed KdV-type DSW. In the RDSW regime, the DSW is in resonance with diffractive radiation, so that it consists of a KdV-type DSW with resonant radiation propagating ahead of it. This resonant radiation is not large enough, however, as to destroy the KdV-type DSW structure, as in the CDSW regime; see Figure 2c. As the DSW in the PDSW and RDSW regimes are perturbed KdV DSWs [36], the solutions in these two regimes can be found using the perturbed KdV DSW solution of Reference [47]. In this previous work, the general eKdV equation, a particular case of which is (42), was asymptotically transformed to the KdV equation, in which the known DSW solution [19; 23] was then used to find the asymptotic DSW solution of the original eKdV equation. This asymptotic DSW solution can be used here based on the eKdV equation reduction (42) of the full nematic equations in the limit \(u_{-}-u_{+}\) small. The work [47] then gives the following PDSW and RDSW nematic DSW solutions. The amplitude of the DSW is  
\[ \begin{array}{l}a=2m(u_{i}-u_{+})+\frac{1}{3}(u_{i}-u_{+})^{2}\{m(1-m)C_{1} +m(m-2)C_{2}\\ +mC_{3}+2m(8-3m)C_{4}\},\end{array}  \tag{45}\]  
Figure 3: Numerical existence regions for nematic DSW types in terms of the nonlocality parameter \(v\) as the initial level \(u_{+}\) varies. (**a**) Full existence regions from highly nonlocal \(v=200\) to local \(v=0\), (**b**) Detail of (**a**) for the transition to an NLS-type DSW. Theoretical boundary (72) between RDSW and CDSW regimes: black (dot-dash) line; theoretical boundary \(U_{s}=c_{\delta}\), (95), between CDSW and TDSW regimes: red (solid) line; theoretical boundary (43) between KdV-type and NLS-type DSW: green (dashed) line. Here, \(u_{-}=1.0\) and \(q=2\). its wavenumber is  
\[ \begin{split} k&=\frac{\pi\sqrt{2\{u_{i}-u_{+}\}}}{K(m) \sqrt{u_{r}}\sqrt{\frac{i}{4}-\frac{d}{4\omega_{r}^{2}}}}\Big{\{}1+\frac{(u_{i} -u_{+})}{12}C_{1}+\frac{(u_{i}-u_{+})(4m^{2}-8m+3)}{12}C_{3}\\ &-\frac{(u_{i}-u_{+})(8m^{2}-14m+11)}{3}C_{4}\Big{\}},\end{split}  \tag{46}\]  
and its mean level is  
\[ \begin{split}&|\tilde{u}|=2u_{+}-u_{i}+\big{(}u_{i}-u_{+}\big{)} \Big{\{}m+\frac{2E(m)}{K(m)}\Big{\}}\\ &-(u_{i}-u_{+})^{2}C_{1}\Big{\{}\frac{3m^{2}K(m)+4 mE(m)-5 mK(m)-2E( m)+2K(m)}{18K(m)}\Big{\}}\\ &-(u_{i}-u_{+})^{2}C_{3}\Big{\{}\frac{(2-m-m^{2})K^{2}(m)+(4m-10) E(m)K(m)+8E^{2}(m)}{6K(m)}\Big{\}}\\ &+2(u_{i}-u_{+})^{2}C_{4}\Big{\{}\frac{(m^{2}-7m+6)K^{2}(m)+2(6m- 11)E(m)K(m)+16E^{2}(m)}{3K^{2}(m)}\Big{\}}.\end{split}  \tag{47}\]  
Here, \(K(m)\) and \(E(m)\) are complete elliptic integrals of the first and second kinds of modulus squared \(m\), respectively. The modulus squared \(m\) is a parameter in these amplitude, wavelength, and mean height expressions. It is determined in terms of the simple wave (similarity) variable \(x/z\) by  
\[\begin{split}&\frac{x}{z}=\frac{\sqrt{2}}{\sqrt{4}}u_{++}+\frac{ \sqrt{2}(u_{i}-u_{+})}{\sqrt{4}}\Big{\{}1+m-\frac{2m(1-m)K(m)}{E(m)+(m-1)K(m)} \Big{\}}\\ &+\frac{(u_{i}-u_{+})^{2}}{3\sqrt{2}q}C_{1}\Big{\{}1+m-\frac{2m( 1-m)K(m)}{E(m)+(m-1)K(m)}\Big{\}}\\ &+\frac{(u_{i}-u_{+})^{2}}{3\sqrt{2}q}C_{3}\Big{\{}2m-1-m^{2}- \Big{(}1+m-\frac{2m(1-m)K(m)}{E(m)+(m-1)K(m)}\Big{)} \tag{48}\]  
This expression for \(x/z\) derives from the characteristic of the KdV modulation equations on which the simple wave DSW solution occurs [23; 24; 47]. The coefficients \(C_{j}(j=1,2,3,4)\) in the above solutions are connected to \(c_{j}(j=1,2,3,4)\) through the relations  
\[C_{1}=\sqrt{2q}c_{1},\quad C_{2}=\frac{8\sqrt{2}q^{3/2}u_{+}}{(4u_{+}^{2}v-q^ {2})}c_{2},\quad C_{3}=\frac{8\sqrt{2}q^{3/2}u_{+}}{(4u_{+}^{2}v-q^{2})}c_{3}, \quad C_{4}=\frac{64\sqrt{2}q^{5/2}u_{+}^{2}}{(4u_{+}^{2}v-q^{2})^{2}}c_{4}. \tag{49}\]  
At the leading, solitary wave edge of the DSW \(m\to 1\) and at the trailing, harmonic wave edge of the DSW \(m\to 0\).