# Nematic Dispersive Shock Waves from Nonlocal to Local
## 4 PDSW and RDSW Regimes
This mean term arises on integrating \(v=\psi_{x}\) to obtain the dispersion relation for \(\psi\) from that for \(v\)[35]. In the limit \(vk^{2}\) large, this dispersion relation becomes (19)  
\[\omega=k\vartheta+\frac{1}{2}k^{2}+\frac{2\bar{\varrho}}{q}+\dots, \tag{52}\]  
again with the carrier wave phase shift \(2\bar{\varrho}/q\) added to obtain the dispersion relation for \(\psi\). In the RDSW regime, the resonant wavetrain has low amplitude; see Figure 2b. In addition, this wavetrain has high frequency relative to the DSW; again, see Figure 2b. So, the appropriate dispersion relation for the resonant wavetrain is (52). In the RDSW regime, the resonance condition for the resonant wavetrain ahead of the DSW has been set by matching the phase velocity of the resonant wavetrain to the velocity of the lead wave of the DSW [34; 35; 36], giving  
\[s_{+}=c_{r},\quad c=\bar{v}+\frac{1}{2}k+\frac{2\bar{\varrho}}{qk}, \tag{53}\]  
so that  
\[k_{r}=s_{+}+\left[s_{+}^{2}-\frac{4}{q}u_{+}^{2}\right]^{1/2} \tag{54}\]  
on setting \(\bar{\varrho}=\sqrt{u_{+}}\) and \(\vartheta_{+}=0\) as the resonant wavetrain propagates on the level ahead. This gives the wavenumber of the resonant wavetrain based on this criterion. The resonant wavetrain then exists if \(s_{+}\geq 2u_{+}/\sqrt{q}\), so that the borderline between the PDSW and RDSW regimes is \(s_{+}=2u_{+}/\sqrt{q}\). Previous work [35; 36] has shown that this theoretical borderline is in excellent agreement with numerical solutions in the high nonlocality limit \(\
u\) large. For fixed \(u_{+}\), as \(\
u\) decreases, a PDSW changes to an RDSW, then to a CDSW; see Figure 3. For instance, for \(u_{+}=0.8\), the DSW changes from PDSW to RDSW at \(\
u=88\), then to CDSW at \(\
u=34\). However, the resonance condition (53), or (54), gives that the DSW changes from PDSW to RDSW at \(\
u=3\), which is the TDSW regime according to \[C_{1}=\sqrt{2q}c_{1},\quad C_{2}=\frac{8\sqrt{2}q^{3/2}u_{+}}{(4u_{+}^{2}v-q^ {2})}c_{2},\quad C_{3}=\frac{8\sqrt{2}q^{3/2}u_{+}}{(4u_{+}^{2}v-q^{2})}c_{3}, \quad C_{4}=\frac{64\sqrt{2}q^{5/2}u_{+}^{2}}{(4u_{+}^{2}v-q^{2})^{2}}c_{4}. \tag{49}\]0 This resonance condition is based on the limit \(\
u k^{2}\gg 1\), but, even if the full dispersion relation (51) were used for the resonance condition (53), the predicted PDSW/RDSW borderline is \(\
u=2.38\), which is still far from the numerical value and close to that for \(\
u k^{2}\gg 1\). The resonance condition (53) is based on resonance between the lead wave of the DSW and diffractive radiation. However, as pointed out previously [36], a DSW is modulated periodic wave so that all waves of the DSW can resonant with diffractive radiation, not just the lead wave, as seen in Figure 5 for a PDSW. Internal resonance will be discussed in detail in Section 5. The phase velocity of a component wave of the DSW is (72). Equating this bore component phase velocity with the nematic diffractive radiation phase velocity determined from the dispersion relation (18) determines the internal resonance. However, even using this internal resonance does not result in a borderline between the PDSW and RDSW regimes in any reasonable accord with numerical solutions; see \[C_{1}=\sqrt{2q}c_{1},\quad C_{2}=\frac{8\sqrt{2}q^{3/2}u_{+}}{(4u_{+}^{2}v-q^ {2})}c_{2},\quad C_{3}=\frac{8\sqrt{2}q^{3/2}u_{+}}{(4u_{+}^{2}v-q^{2})}c_{3}, \quad C_{4}=\frac{64\sqrt{2}q^{5/2}u_{+}^{2}}{(4u_{+}^{2}v-q^{2})^{2}}c_{4}. \tag{49}\]1 A resonant wavetrain will then exist if the internally resonant waves can propagate out of the DSW, that is their group velocity is greater than the velocity of the lead solitary wave of the DSW. However, even this condition does not give the correct boundary between the PDSW and RDSW regimes as the nonlocality parameter \(\
u\) decreases. The issue of internal resonance and its relation to the existence of the PDSW and RDSW regimes merits further study. In this regard, the recent work [55] on the interaction of linear wavepackets and DSWs could be relevant.