# Nematic Dispersive Shock Waves from Nonlocal to Local
## 2 Nematic Equations
* RDSW (radiating DSW): As the nonlocality \(v\) decreases, the DSW becomes resonant with a resonant wavetrain attached to the leading edge of the DSW, as illustrated in Figure 2b. As all individual waves in the DSW are resonant, resonant waves are emitted from the DSW, which results in the DSW not being rank ordered [36]. * CDSW (crossover DSW): As the nonlocality decreases and the amplitude of the resonant wavetrain grows, the DSW becomes unstable with a total loss of the rank ordering of the waves of the DSW [26; 36]. This regime is illustrated in Figure 2c. * TDSW (traveling DSW): As the amplitude of the resonant wavetrain grows, the shedding of conserved quantities into resonant radiation eventually destroys the DSW, leaving a high amplitude resonant wavetrain with a negative polarity solitary wave linking this wavetrain to the intermediate level [26] as seen in Figure 2d. While there is a solitary wave linking the resonant wavetrain to the intermediate level, this linking can be conveniently treated as a Whitham shock [48], a shock wave in the Whitham modulation equation variables. The resonant wavetrain is brought down to the level \(u_{+}\) ahead by a partial DSW [36]. This partial DSW has linear dispersive waves at its leading edge but has a finite wavelength wave at its trailing edge, that is, it is not bounded by solitary waves at the trailing edge [38; 49], as for a standard DSW. * RNLS DSW (radiating NLS DSW): Further decrease in the nonlocality results in the amplitude of the linking solitary wave becoming negligible, so that the wave form consists of a (stable) resonant wavetrain headed by a partial DSW which brings the wavetrain down to the level \(u_{+}\) ahead, as seen in Figure 2e. This DSW regime does not occur in the high nonlocality limit as it is a \"bridge\" to the local NLS DSW for \(v=0\)* NLS DSW: As the nonlocality \(\
u\) decreases to \(\
u=0\) and the nematic Equations (11) and (12) reduce to the NLS equation, the resonant wavetrain length contracts and the leading partial DSW evolves to a full DSW, with linear dispersive waves at the leading edge and solitary waves at the trailing edge. The resonant wavetrain then disappears and the leading DSW bringing the wavetrain down to the level \(u_{+}\) ahead attaches to the intermediate level with \(|u|=u_{i}\), as in Figure 2f. The resulting DSW is essentially an NLS DSW, which completes the transition from the KdV-type DSW for high nonlocality, that is for low power beams, to a NLS DSW for low nonlocality, that is for high power beams. In addition to these six DSW types, when \(u_{-}-u_{+}\) is large enough as \(u_{+}\to 0\), there is an additional DSW type which is a sub-case of the TDSW regime, the vacuum DSW (VDSW) [37], for which the electric field \(u\) of the resonant radiation vanishes at a point [35,36]. As the solution for this DSW type has been previously analyzed [36], and it is a sub-case of the TDSW regime, it will not be considered here. In addition,  
Figure 2: Numerical solutions of nematic Equations (11) and (12) for initial condition (13) with \(u_{+}=0.8\) and \(u_{-}=1.0\). Red (solid) lines: \(|u|\) at \(z=1000\); green (dashed) lines: \(\phi\) at \(z=1000\); blue (dotted) lines: \(|u|\) at \(z=0\) (upper) and \(\phi\) at \(z=0\) (lower). **(a)** PDSW with \(\
u=200\), **(b)** RDSW with \(\
u=40\), **(c)** CDSW with \(\
u=10\), **(d)** TDSW with \(\
u=3.0\), **(e)** resonant NLS type DSW with \(\
u=1.0\), **(f)** NLS type DSW with \(\
u=0.5\). Here, \(q=2\). the analytical work of this manuscript is based on \(u_{-}-u_{+}\) being small, which is not valid in the VDSW regime. Table 1 shows that the transition regimes of the TDSW and RNLS DSWs which evolve the DSW from the KdV-type DSW for high nonlocality to the NLS-type DSW for weak nonlocality exist for very restricted ranges of \(\
u\), so that this transition is rapid. Over most of the range of \(\
u\) the DSW is of KdV-type, one of the PDSW, RDSW, and CDSW types.