# Nematic Dispersive Shock Waves from Nonlocal to Local
## 8 Conclusions  
The structure of the nematic DSW (dispersive shock wave) solution of the defocusing nematic equations governing the propagation of an optical beam through a cell filled with nematic liquid crystals has been investigated using a combination of numerical solutions of the equations governing the beam, consisting of an NLS-type equation for the optical beam and an elliptic medium response equation, and solutions of the governing nematic equations using Whitham modulation theory and/or asymptotic solutions. In contrast to previous work [34; 35; 36], the evolution of the DSW structure was studied as the power of the optical beam varied, from the experimental low power for which the nematic response is nonlocal to high power for which the response is local. As the beam power varies, it was found that the nematic DSW transitions between six regimes, four of which were studied in previous work [34; 35; 36]. The two NLS-type DSWs do not exist in the low power regime studied in this previous work. However, the experimental verification of these high power DSW types is questionable as the powers required for their existence are unrealistic due to the possible excessive medium heating the high beam powers would cause. Excellent agreement was found between numerical solutions and analytical solutions for the four physically relevant DSW regimes, the PDSW, RDSW, CDSW, and TDSW regimes displayed in Figure 2a-d. In particular, the analytical theory gives good agreement for the boundaries between the existence regions for five of the DSW types as the nonlocality parameter \(\
u\) varies, the exception being the boundary between the PDSW and RDSW regimes.  
It has been found that the details of the nematic DSW, for instance, its lead wave amplitude and velocity and the amplitude of the associated resonant radiation, are well approximated by the nematic equations in the high nonlocality limit, the nonlocality parameter \(\
u\) large, as the DSW transitions to the local limit. This holds for the PDSW, RDSW, CDSW and TDSW regimes with the nonlocality parameter \(\
u\) ranging from \(O(100)\) to \(O(1)\). As the analysis of the nematic DSW is much easier in the high nonlocality limit based on asymptotic analyses with \(\
u\gg 1\), this is an important result for future analysis of the nematic DSW in its various regimes and over its nonlocality range.  
There are still a number of issues which could be addressed by future work. An outstanding issue is the correct determination of resonance between the nematic DSW and diffractive radiation. In contrast to the high nonlocality limit with \(\
u\) large [34; 35; 36], as \(\
u\) decreases from the high to the low nonlocality limit, the beam power increases, and the resonance condition used in previous work in the RDSW and CDSW regimes that the velocity of the lead solitary wave of the DSW matches the linear phase velocity of the resonant waves does not agree with numerical solutions. The theoretical transition from the PDSW regime for high nonlocality to the RDSW regime due to the onset of resonance as \(\
u\) decreases occurs for \(\
u=O(1)\), while numerical solutions show the transition for \(\
u=O(100)\) to \(O(10)\). As noted in previous work [36], and in contrast with other work on resonant DSWs, not only is the lead wave of the DSW in resonance with diffractive radiation, but the whole modulated periodic wave which forms the DSW is in resonance. However, even this observation does not yield the correct resonance condition for the RDSW regime as the nonlocality parameter \(\
u\) decreases below the highly nonlocal limit. The correct resonance condition between the DSW and diffractive radiation requires further study. This should be an important general issue for all resonant DSWs beyond the specific application to nematic liquid crystals.  
This research received no external funding.  
The authors declare no conflict of interest.