# Nematic Dispersive Shock Waves from Nonlocal to Local
## 1 Introduction
These modulation equations form a hyperbolic system, so that the cnoidal wave solution of the KdV equation is modulationally stable. It was subsequently realized that a simple wave solution of the KdV modulation equations is a DSW [23], even though the initial condition is a step, which is not slowly varying. This DSW solution is in excellent agreement with numerical solutions of the KdV equation [24]. The key to the determination of the simple wave DSW solution is the ability to set the modulation equations in Riemann invariant form. If the nonlinear dispersive wave equation governing the DSW is integrable, then its Whitham modulation equations can be automatically set in Riemann invariant form [25], so that the DSW solution can easily be found. As mentioned above, the standard DSW form, termed of KdV type [19], is a modulated periodic wave with solitary waves at one edge and linear dispersive waves at the other. A non-standard DSW type is a resonant DSW [26; 27], for which the waves of the DSW are in resonance with (linear) dispersive waves, resulting in a resonant wavetrain being emitted from the DSW. Resonant DSWs also occur for the KdV equation with next higher-order dispersion, i.e., fifth-order dispersion, namely for the Kawahara equation [28], and the NLS equation with next order, third-order dispersion [29; 30; 31; 32]. If the emitted resonant wavetrain is of large enough amplitude, the KdV-type DSW structure can be destroyed; this results in the so-called traveling dispersive shock wave (TDSW) regime [26; 27], consisting of a resonant wavetrain with a negative polarity solitary wave, which is the remnant of the DSW, linking this to the level behind [26]-- see Figure 2d below for an example of such a TDSW. A nematic liquid crystal is a focusing medium; thus, optical waves are modulationally unstable; as a result, an optical DSW is not supported. However, the addition of azo dyes to the nematic medium changes its response so that it becomes defocusing [33]; in this case, nematics can support DSWs [34; 35; 36]. A nematic DSW is an example of a resonant DSW [34; 35; 36]. In these works, the nematic DSW was studied in the highly nonlocal limit, for which the nematic elastic response extends far beyond the light beam, with the nematic DSW generated by a step jump in the optical intensity. While the nematic equations are of NLS-type [2], in the highly nonlocal limit the nematic bore is of KdV-type and is well described by the DSW solution of the KdV equation. The nematic DSW structure is highly dependent on the size of the jump of the optical electric field intensity generating it, with six distinct DSW types identified [36]. As stated above, in the highly nonlocal limit the nematic DSW is of KdV-type with the DSW having positive polarity. However, in the limit of weak nonlocality, the nematic equations reduce to the NLS equation [2,4], and the nematic DSW becomes the NLS DSW, which is non-resonant. The degree of nonlocality of the optical response of a nematic is inversely proportional to the power of the optical beam, with the response being highly nonlocal for lower power beams, transitioning to local as the beam power increases [2,4], as will be detailed in Section 2. In this work, the evolution of the nematic DSW structure as the degree of nonlocality ranges from highly nonlocal (low power beams) [34-36] to local (high power beams) will be studied. As the nonlocality decreases, the changes in the DSW structure from those previously found [36] in the limit of high nonlocality to the standard NLS DSW [37] will be identified, and the solutions for these will be derived. It is found that there exist two additional DSW regimes over those for large nonlocality, including the NLS DSW for zero nonlocality. The new DSW regime is a transition between the KdV DSW behavior for large nonlocality and the NLS DSW behavior for very small nonlocality.