# Nematic Dispersive Shock Waves from Nonlocal to Local
## 1 Introduction  
Nematic liquid crystals form an ideal medium to study nonlinear optics due to their \"huge\" nonlinearity, which is orders of magnitude larger than that of optical fibers, so that nonlinear effects can be observed over millimeter distances [1, 2, 3, 4]. In particular, the refractive index of nematic liquid crystals increases with optical intensity, so that they form a focusing medium. When a light beam propagates through a nematic liquid crystal, the electric field of the electromagnetic wave induces dipoles in the nematic molecules, which then rotate, changing the refractive index. In addition, nematic liquid crystals have a nonlocal response to an optical beam in that the elastic response of the nematic extends far beyond the optical forcing [4]. An optical beam propagating through a nematic medium can then form its own waveguide, resulting in a self-guided beam, an optical solitary wave, termed a \"nematicon\" [2, 3, 5, 6], which was first experimentally generated and observed in 2000 [6]. Since this first observation, nematicons, and related solitary-type waves, such as optical vortices, have become a theme of intense experimental and theoretical research effort, driven both by interest in the nonlinear optics of nematic liquid crystals and also by their potential applications in optical devices [7, 8, 9, 10]; see Refs. [2, 3, 5, 11] for general reviews on the nonlinear optics of nematic liquid crystals. Solitary waves are generic wave forms for nonlinear dispersive wave equations [12], first observed and studied in the context of water waves [12, 13] and fluid dynamics [12], but are widespread in nature arising, e.g., in solid mechanics [14], biology [15], ecology [16],and the aforementioned context of nonlinear optics [2; 17; 18], for instance. One of the appealing features of solitary waves, in addition to their widespread occurrence in nature, is that they are localized waves with steady profiles, which makes them easier to study theoretically. In addition to this, many generic nonlinear dispersive wave equations, such as the Korteweg-de Vries (KdV), nonlinear Schrodinger (NLS), and Sine-Gordon equations, are completely integrable systems via the Inverse Scattering Transform method [12; 18]. Thus, a general initial condition for these equations will form a finite number of solitary waves, plus dispersive radiation. In addition, solitary wave solutions of integrable nonlinear dispersive wave equations, solitons, interact \"elastically\", i.e., they emerge unscathed out of the interaction without any change in their form; hence, due to this particle-like behavior, solitary waves are termed solitons for such equations. Another generic wave form supported by nonlinear dispersive wave equations are dispersive shock waves (DSWs), also termed undular bores; these structures are as widespread in nature as solitary waves, with well known examples being tidal bores and tsunamis [19]. In contrast to a solitary wave, a DSW is a non-steady wave form which continuously expands. A DSW is a dispersive regularization of a discontinuity and is a modulated periodic wavetrain with solitary waves at one edge and linear, dispersive waves at the other--see Reference [19] for a general review of DSWs. Since DSWs are non-steady waveforms, their study is more difficult than that for solitary waves. The development of DSW solutions of nonlinear dispersive wave equations relies chiefly on Whitham modulation theory [12; 20; 21; 22], which is a version of the asymptotic method of multiple scales that is used to analyze slowly varying periodic wavetrains. Whitham modulation equations are a system of partial differential equations which govern the parameters of a slowly varying wavetrain, such as its amplitude, wavenumber, frequency, and mean height. If this system is hyperbolic, then the underlying wavetrain is modulationally stable, while, if it is elliptic, the wavetrain is unstable [12]. A major achievement of Whitham modulation theory was the development of the modulation equations for the KdV equation [12; 21].