Chapter 16: Dispersive Shock Waves in Reorientational and Other Optical Media
School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, New South Wales, Australia
16.1 Introduction
Nematic liquid crystals are an example of a nonlinear nonlocal optical medium in which rod-like molecules orientate in response to a light beam, or some other applied electric fields. This is because the electric field induces dipoles in the organic molecules. Nematic liquid crystals have received much attention from researchers, as nonlinear effects can be observed over small length scales. A series of elegant experiments have shown that stable optical spatial solitary waves, so-called nematicons, can propagate in nematic liquid crystals [1, 2]. Such solitary waves have generated much interest because of their possible applications as reconfigurable “circuits” for all-optical information processing [3–5].
Nematic liquid crystals are just one example [6] in a variety of optical dielectrics that support spatial solitary waves (Chapter 1). Other optical media include photorefractives [7], quadratic media [8], glasses [9], thermal media [10], and colloids [11]. In all these media, solitary waves occur because of a balance between diffraction (in the spatial domain) or dispersion (in the temporal domain) and nonlinear (self-) focusing. In all cases, a nonlinear interaction between the medium and the light alters the refractive index of the medium.
The nonlinear Schrödinger (NLS) equation is central to our theoretical understanding of nonlinear optics and the field of fiber-based telecommunications. In this context, the NLS equation was first derived in Reference 12 to describe the evolution of the slowly varying envelope of an optical pulse. Derived asymptotically from Maxwell's equations, it assumes slow variation in the carrier envelope and that the refractive index is linearly dependent on the optical intensity (the Kerr dependence). The NLS soliton solution occurs because of a balance between dispersion and nonlinear effects. As the NLS equation is integrable, the NLS solitons undergo elastic collisions and a vast array of analytical solution techniques are available for its analysis, such as the inverse scattering transform. Nonlocal effects are important for some of these optical applications, such as nematic liquid crystals, and the governing equations then become a coupled Schrödinger–Poisson system.
For nonlinear dispersive (or diffractive) wave equations, initial discontinuities are resolved through dispersion into a dispersive shock wave (DSW). Also termed as an undular bore or a collisionless shock, a DSW is a modulated wave train consisting of solitary waves at its leading edge and linear waves at its trailing edge. The DSW then gives a smooth transition across the discontinuity. Mathematically DSW solutions are derived using Whitham's modulation theory [13], who first applied the technique to the Korteweg–de Vries (KdV) equation. Modulation theory develops partial differential equations, which describe the evolution of a slowly varying wave train. In general, if these modulation equations are hyperbolic, the wave train is modulationally stable and the DSW is found as a simple wave solution. If the modulation equations are elliptic, the wave train suffers modulational instability (MI).
The Schrödinger–Poisson system can be of defocusing (stable DSW) or focusing (unstable DSW) type, depending on the properties of the physical medium. The type depends on whether the refractive index of the medium increases or decreases with increasing light intensity. Reorientational nematic liquid crystals tend to be focusing (Chapter 1) whereas photorefractives, Bose–Einstein condensates, and thermal media can be of either type. Hence, a vast array of optical media with different properties are available for experimentation and investigation by researchers. DSW for focusing NLS-type equations are subject to MI and do not persist at long length or timescales. However, recent experiments show that the DSW do occur for focusing media. In Reference 14, it has been shown that a DSW develops for a focusing photorefractive medium; the MI can be suppressed by using partially incoherent light. Moreover, the development of DSW is also observable in nematic liquid crystals; in References [15–17], MI was studied in nematic liquid crystals over typical experimental length scales.
This chapter reviews the existing experimental, numerical, and theoretical results for the development of DSW in nematic liquid crystals and a range of other optical media and also analyze in detail a method to obtain approximate analytical results for DSW governed by focusing-type equations.
16.2 Governing Equations and Modulational Instability
A coupled Schrödinger–Poisson system
is an appropriate model for optical waves in a nonlinear nonlocal medium. The scalings in Equation 16.1 are chosen to be consistent with Reference 18. E is the electric field. In nematic liquid crystals, the angle θ is the deviation of the molecular director from its pretilt. Even though in other optical applications, θ can have other physical meanings, we term θ the director orientation. ν is a measure of the nonlocality and q is used to classify the equation type; q = 1 is the focusing case and q = − 1 is the defocusing case. The Laplacian ∇2 is in the (x, y) plane and the governing equations support (2 + 1)D spatial solitary waves with the z coordinate a time-like variable. The physical meaning of θ in thermal media is temperature, whereas in colloidal media it is particle density. In photorefractive materials, nonlocal effects are due to charge transport, whereas, for Bose–Einstein condensates, they are caused by longrange particle interactions.
In the local limit, where ν = 0, θ = |E|2, and the NLS equation
16.2 
is obtained. Numerical and analytical results for photorefractive crystals and Bose–Einstein condensates are typically considered in the local limit.
A model related to Equation 16.1 is the nonlocal NLS equation
for various response functions R [19–21]. Solutions of Equation 16.1 in (2 + 1)D correspond to a response function R, given in terms of modified Bessel functions, which results in solutions of the nonlocal NLS equation (Eq. 16.3) being difficult to find. As a consequence, most of the analysis of this nonlocal NLS equation has been done for simplified response functions, such as the Gaussian and exponential types.
In the case of (1 + 1)D, the governing equations (Eqs 16.1 and 16.3) are equivalent. In this case, the director equation, the second of Equation 16.1, can be solved using Green's functions and the result substituted into the first of Equation 16.1, resulting in the nonlocal NLS equation
where 
. It is possible to obtain approximate solitary wave solutions to Equation 16.4 in terms of Bessel functions, in the limit of large ν (or θ small) [22].
The continuous wave (cw) solution for Equation 16.1 is
16.5 
The cw for the defocusing case is stable, so we consider the focusing case with q = 1. The electric field and director angle are perturbed by small quantities
16.6 
A modal expansion for the perturbation of the form
16.7 
is used, which is then substituted into Equation 16.1, giving
16.8 
Hence, the cw is unstable if
This stability result is the same, suitably rescaled, as Equation 39 in Reference 23. It implies that the cw is subject to MI developing from perturbations of small wave number or long wavelengths. In the local limit, ν → 0 the NLS instability band 
is obtained [4, 13]. In the nonlocal limit, ν → ∞ the instability band (Eq. 16.9) becomes 
with a maximum rate of instability of 
. This result shows that increased nonlocality suppresses MI as the instability band and the maximum growth rate are both reduced.
An appropriate initial condition for the development of a DSW in (1 + 1)D is
at z = 0. To the left and right of the initial shock, two background cw with different amplitudes occur. This initial condition is experimentally feasible [14], who created an initial step condition by blocking half the initial light beam using a razor blade, which implies an initial condition of the form (Eq. 16.10) with bm = 0.
16.3 Existing Experimental and Numerical Results
In Reference [23], the development of MI in a (1 + 1)D nonlocal NLS equation (Eq. 16.3) has been considered. A stability analysis showed that MI always occurs for the focusing case, whereas in the defocusing case, the occurrence of MI is dependent on the particular form of the response function. Three main types of response kernel were considered: the exponential function, which corresponds to the (1 + 1)D version of Equation 16.1, and the Gaussian and rectangular response kernels. The MI gain profiles showed that the maximum gain decreases as the degree of nonlocality increases, which confirms that nonlocality can suppress the development of MI but cannot eliminate it. A modulational transverse stability analysis of long, one-dimensional beams for the (focusing) nematicon equations was undertaken in Reference 24. Again, nonlocality was found to suppress MI.
In Reference 15, the development of MI for nematic liquid crystals has been considered. The experimental results showed that a wide input beam develops a periodic pattern, as the propagation distance increases, which is progressively amplified. Analysis of the FFT spectrum of intensity showed that the maximum gain corresponds well to the predictions of a stability analysis. From a practical viewpoint, the z location at which MI occurs in this and other experimental scenarios is of great interest, as it determines whether or not DSW can be observed in focusing media. Figure 3 in Reference 15 showed that MI develops at z ≈ 300 μm for an input beam with power P = 352 mW. [This power level is very high for experiments involving nematicons, with a power level of P = O(10) mW much more typical; see References 1, 25, and 26. The reason that such a high power level was used was that MI was not observable, over typical experimental length scales, using lower power levels.] In Reference 15 for the MI gain λ, the relation 
is demonstrated, which implies that the onset of MI will occur at millimeters length scales for typical power levels.
In Reference 27, the development of DSW in nonlocal media has been considered, both in the focusing and defocusing cases. In the focusing case, it was shown that nonlocality suppresses the development of MI, so that the development of a DSW could be seen for a certain length scale, before MI destroys the bore structure. The defocusing case is modulationally stable, so that the shock develops into a stable DSW consisting of dark solitary waves. Experimental results were also presented for the development of a shock in a defocusing thermal medium, with good qualitative comparisons obtained with numerical results. In Reference 28, the generation of a DSW in a defocusing thermal medium has been investigated, finding a good qualitative comparison between their experimental and numerical results. In Reference 29, the transverse stability of dark solitary waves and DSW in a nonlocal defocusing medium has been theoretically examined and it has been shown that the nonlocal response has a strong stabilizing effect.
In Reference 30, experimental results for a defocusing photorefractive crystal have been carried out, where a bright hump was imposed on a uniform low-intensity background. Results were presented for the development of DSWs with both line and circular geometries. The DSWs are qualitatively similar to the development of undular bores for the KdV equation and other nonlinear wave equations. In Reference 10, a defocusing thermal medium was considered, which has significant nonlocal effects. An initial condition of a Gaussian hump was used, and a two-dimensional DSW with circular geometry was obtained. The experimental results were qualitatively similar to those for the photorefractive medium. Numerical results for a nonlocal version of the NLS equation, with a Gaussian response function, were also presented. It was found that the effect of nonlocality is to shorten the length of the DSW. In References 14 and 31, DSW development in a focusing photorefractive media was considered. Partially spatially incoherent light is used to suppress MI and the development of a 1D DSW is captured. The shock width was found to be a linear function of nonlinearity and coherence length.
In Reference 32, a generalized focusing NLS equation with three additional terms representing third-order dispersion, self-steepening, and intrapulse stimulated Raman scattering (SRS) has been considered. Authors found an analytical kink solution of this equation that resolves an optical shock in a smooth, monotonic manner with no oscillatory waves. This kink solution is a viscous bore solution, with the SRS giving the required loss, as opposed to the oscillatory, and frictionless, DSW. Authors further considered the numerical evolution of pulses composed of kink–antikink pairs and a super-Gaussian profile and found that there is a redshift in the pulse spectra, which is typical of SRS. In Reference 33, a NLS equation with an SRS term has been considered as well, finding out a kink (viscous bore) solution, which generalized one found in Reference 32. This kink solution could be either monotone or oscillatory, depending on the parameter choices. Lastly, the kink was found to be unstable in both the focusing and defocusing cases.
16.4 Analytical Solutions for Defocusing Equations
Few analytical solutions are available for DSW in nonlocal defocusing media. In the local limit, the Schrödinger–Poisson equation becomes the NLS equation and exact solutions for DSW are then available. The analytical results for the defocusing NLS equation, and related local NLS-type equations, give a useful qualitative indication of the likely nature of DSW in nonlocal media.
In Reference 34 the evolution of a shock for the defocusing NLS equation has been considered. In general, the evolution of an initial discontinuity leads to the development of two waves separated by a plateau region. The two waves can be either a DSW or a rarefaction wave, which is nonoscillatory. The occurrence of a vacuum point, at which the light intensity is zero at a given location in the DSW, was also analyzed. In Reference 35, a method to obtain analytical results for dispersive wave equations has been developed, which are defocusing in nature but not integrable. The method allows many analytical features to be obtained from the hyperbolic modulation equations, such as the amplitude of the lead solitary wave and the width of the DSW region.
In Reference 36, the development of optical shocks governed by a defocusing NLS equation with saturable nonlinearity has been considered. Analytical approximations were found for the rarefaction wave and also the leading and trailing edges of the DSW. Figure 16.1 shows the numerical evolution of a DSW for the defocusing NLS equation with saturable nonlinearity, illustrating the complexity of DSW in defocusing media. A plateau region of constant amplitude has formed, and two DSW are developing on either side of the plateau region. One DSW matches the plateau region to the mean level on the left of the Figure, ρ = 10, whereas the other DSW matches the plateau region to the mean level on the right, ρ = 1. Moreover, a vacuum point, where ρ = 0, has formed in the DSW on the right.
Figure 16.1 The light intensity ρ versus x. Shown are numerical solutions from the defocusing NLS equation with saturable nonlinearity. Source: The figure is from Reference 36.
In Reference 37, the diffraction of light propagating past a reflecting wire in a photorefractive media has been considered. The governing equation was a defocusing NLS equation with saturable nonlinearity and a potential term representing the obstacle. Analytical solutions were developed to describe the optical “ship waves” generated at the obstacle.
For particular choices of initial conditions, a single DSW occurs as a solution of the defocusing NLS equation, and in Reference 34 a simple wave solution to describe this case has been found. The hydrodynamic representation is used where
16.11 
and ρ and v are analogs of the hydrodynamic density and velocity, respectively. The initial conditions considered were
and ρ0 < 1, so a step down in amplitude is considered. The defocusing NLS equation has a traveling wave solution defined in terms of cnoidal functions (see Equation 4 in Reference 34), which is a function of the modulus m. The two limits of the cnoidal wave solution are m = 1, which is the soliton solution, and m = 0, which is a linear wave. The simple wave solution is an expansive fan of characteristics with the other Riemann invariants remaining constant throughout the bore. The bore solution is
E, K, and Π are complete elliptic functions of the first, second, and third kinds, respectively, 
is the mean light intensity, 
is the mean velocity whereas the peak and trough envelopes to the DSW are given by 
and 
, respectively. The solution is completed by the formula for the characteristics in the fan, given by
Equations 16.13 and 16.14 represent the analytical DSW for the defocusing NLS equation. For ρ0 < 0.25, a vacuum point occurs in the DSW, which means that the amplitude ρ = 0 at a particular location in the bore.
16.5 Analytical Solutions for Focusing Equations
The usual method to find the DSW of nonlinear wave equations is to construct the associated modulation equations, and if these equations are hyperbolic, find the DSW as a simple wave solution [38]. The modulation equations for the focusing version of Equation 16.1 are elliptic, hence subject to MI, and so do not possess a simple wave solution. However, a number of experimental results show that a DSW forms in focusing media, for a certain length scale, z, before MI dominates and the bore structure is destroyed. These emerging experimental results are a strong motivation for the development of analytical results to describe DSW in focusing media.
In Reference 39, the modulation equations for the focusing NLS equation have been considered and an analog of the simple wave solution has been found. This self-similar solution describes the decay of a uniform initial condition, which is perturbed at a given point. An oscillatory structure develops with solitons in the central region (at the location of the initial perturbation) and linear waves at the edge of the disturbance. Analytical results for the soliton amplitude and the width of the oscillatory region are found and discussed, as is the characteristic time for the development of instabilities. In Reference 40, modulation equations for the Landau–Lifshitz equation have been developed, which describes ferromagnetic materials. An analytical oscillatory solution, which is qualitatively similar to that in Reference 39 has been derived.
As the standard method, based on modulation theory, does not yield results for DSW in focusing media, an alternative method, termed as uniform soliton theory, will be used. This was developed in Reference 41 for the KdV equation and widely applied to other nonlinear wave equations, such as the magma equation (Eq. 16.15). In Reference 42, an initial boundary value problem for magma flow in the Earth's mantle has been considered, which was formulated as
where f is the magma fraction. Physically, a large reservoir of magma is located at x = 0 with liquid fraction fr, and the magma then propagates into the semi-infinite domain, which has background liquid fraction fb. They found modulation equations to describe the evolution of the magma DSW, by assuming slow variations in the traveling wave solution and averaging the two conservation equations for Equation 16.15. Figure 16.2 shows the development of a magma DSW for Equation 16.15. A partial DSW occurs with solitary waves at the leading edge, x ≈ 17, whereas at x = 0, cnoidal waves are generated. As modulation theory gives averaged information about the DSW, the mean level and the wave peak and trough envelopes are plotted from the modulation theory solution. An excellent comparison between the modulation theory and numerical solutions was obtained.
Figure 16.2 The liquid magma fraction f versus x at t = 60. Shown are the modulation theory (dashed lines) and numerical (solid lines) solutions. The modulation theory solution consists of the wave peak and trough envelopes and the mean height. Source: The figure is from Reference 42.
Uniform soliton theory was also successfully applied to the magma evolution problem (Eq. 16.15) in Reference 42. The theory involves using conservation laws of the governing equation and also assuming that solitons of constant amplitude are generated at a constant rate. This then gave a transcendental equation
where
Equation 16.16 implicitly determines the amplitude a of the solitary wave generated by the shock, in terms of fr and fb. Figure 16.3 shows the solitary wave amplitude a versus the background magma fraction level fb. Compared are the solitary wave amplitude at the leading edge of the DSW, as given by the uniform soliton theory, modulation theory, and the numerical solution. It can be seen that the uniform soliton theory gives an excellent comparison with the numerical solutions and outperforms modulation theory as the background magma fraction fb becomes very small. In this limit, as the solution is a train of solitary waves, the averaging process becomes invalid, as the wavelength is infinite, hence the modulation theory breaks down.
Figure 16.3 The solitary wave amplitude a versus liquid magma fraction fb. Shown are the solutions from modulation theory (dashed lines), uniform soliton theory (solid lines), and numerical solutions (squares). Source: The figure is from Reference 42.
16.5.1 The 1 + 1 Dimensional Semianalytical Soliton
The uniform soliton theory discussed in Reference 41 is based on the exact soliton solution of the KdV equation. No such exact solitary wave solution of the Schrödinger–Poisson equations (Eq. 16.1) exists. Therefore, a variational approximation for this solitary wave will be found, which has been found to be accurate in previous work on nematicon evolution [43–45]. The variational approximation to the solitary wave in (1 + 1)D will be used in the uniform soliton theory. Equation 16.1 has the Lagrangian
for q = 1, where the * superscript denotes the complex conjugate. An approximate steady solitary wave solution in (1 + 1)D is sought using a trial function similar to that used by Minzoni et al. [45]
The trial function for the electric field is based on the NLS soliton and has amplitude a and width w. The trial function for the director angle θ has amplitude α and width β and its form is chosen so that the relationship θ = |E|2 can be satisfied in the local limit, as ν → 0. The wave number k is chosen to be the same as the wave number in the cw initial profile. This results in the averaged Lagrangian
16.19 
For β ≠ w, the integral I cannot be evaluated in closed form. The variational equations, which give the solitary wave parameters, are then
16.20 
16.21 
16.22 
16.23 
Variational equations (Eqs 16.20–16.23) are four equations in the five parameters and represent a one-parameter family of solitary waves. In the local limit, when ν = 0, variational equations (Eqs 16.20–16.23) give the exact NLS soliton solution
16.24 
16.5.2 Uniform Soliton Theory
The approximate solitary wave solution (Eq. 16.18) in (1 + 1)D will be used to find the amplitude of the solitary waves, generated in a DSW governed by the Schrödinger–Poisson equations (Eq. 16.1), using the method of [41]. This method is based on assuming that the shock is resolved by diffraction into a train of uniform solitary waves. The amplitude and spacing of these nematicons is calculated by using global conservation laws, which must hold no matter what form of solution is used [41].
The initial condition (Eq. 16.12) is considered with bm = 0 so to the right of the bore the background light intensity is zero. The mass and energy conservation laws of Equation 16.1 in (1 + 1)D are
respectively. These conservation laws are named from invariants of the Lagrangian (Eq. 16.17) [46] and do not correspond to these quantities in optical situations. For instance, the mass conservation law (Eq. 16.25) corresponds to conservation of power in the optical context.
Integrating the conservation equations (Eqs 16.25 and 16.26) over the bore then gives
Assuming that N solitary waves have been generated, the integrals on the left-hand side of Equation 16.28 are N times the integral for one solitary wave. From the trial functions (Eq. 16.18), the mass and energy for a single solitary wave are
Substituting Equation 16.29 into Equations 16.27 and 16.28 and taking the ratio of the two equations gives the transcendental equation
for the amplitude a of the solitary wave. For a given height am of the initial shock and choice of ν, Equations 16.20–16.23 and Equation 16.30 are five equations for the five parameters describing the solitary waves generated at the shock.
In the local limit with ν = 0, the uniform soliton theory has an exact analytical solution given by
16.31 
Hence, in this limit, the lead soliton in the DSW has an electric field amplitude that is 2.45 times the height of the initial shock and the director pulse has an amplitude of 6 times the initial shock (in θ). By way of contrast, for the KdV DSW, the lead soliton is twice the height of the initial shock.
16.5.3 Comparisons with Numerical Solutions
Here, numerical solutions of the focusing version of Equation 16.1 are compared with the results of the uniform soliton theory. Figure 16.4 shows the predictions of uniform soliton theory for Equation 16.1. Shown are the parameters of the solitary wave, a, α, w−1, and β−1 versus ν. The initial shock has an amplitude am = 0.5. At ν = 0, the solitary wave has an electric field amplitude of a = 1.22 and a director angle amplitude α = 1.5. As ν increases, the electric field amplitude a, after a slight dip, increases, whereas the director angle, α, decreases. At ν = 0, the width of both pulses is β = w = 0.57. As ν increases, the widths of both pulses increase, with β, the width of the director pulse, becoming much larger than the width of the electric field pulse w. This is consistent with the nonlocal limit, as for ν → ∞, the director pulse becomes much broader than the electric field.
Figure 16.4 Solitary wave parameters versus ν. Shown is a (upper solid line), α (lower solid line), w−1 (upper dashed line), and β−1 (lower dashed line) from uniform soliton theory. The other parameter is am = 0.5. Source: The figure is from Reference 18.
Figure 16.5 shows the electric field and director angle amplitudes, a and α, versus ν. The other parameters are am = 0.2 and k = 0. Shown are the predictions of uniform soliton theory and the numerical solutions. Two different numerical estimates of the solitary wave amplitudes are shown, the amplitude of the first solitary wave generated by the shock and the maximum solitary wave amplitude in the bore, averaged over a range of z. Averaging is used as there is some oscillation in the solitary wave amplitude as the bore develops, because of the nonlocality. The general trend is that, as ν increases, the amplitude of the electric field increases and that of the director angle decreases. The comparison between the theoretical results and the numerical estimates of the maximum solitary wave amplitude is excellent. For the first solitary wave generated, the uniform soliton theory overestimates the numerical predictions by about 25%.
Figure 16.5 Solitary wave amplitudes versus ν. Shown is a, α (solid lines) from uniform soliton theory. Numerical estimates are for the first wave (circles) and the average maximum amplitude (squares). The other parameters are am = 0.2 and k = 0. Source: The figure is from Reference 18.
Figure 16.6 shows the numerical solutions for |E| and θ versus x at z = 300. The other parameters are ν = 1, am = 0.2, and k = 0. This figure shows a classical DSW, similar to the magma DSW of Figure 16.2. Solitary waves occur at the front of the DSW and linear waves of small amplitude at the rear. The lead wave has amplitude a = 0.4 and α = 0.14. The uniform soliton theory gives a = 0.48 and α = 0.18, which overestimate the numerical values by 20%
and 28%
, respectively. As this example is for the local limit, the director pulse has approximately the same width as the electric field. For this example, MI does not occur until z ≈ 600.
Figure 16.6 Numerical solutions of Equation 16.1 versus x at z = 300. Shown are |E| (solid line) and θ (dashed line). The other parameters are ν = 1, am = 0.2, and k = 0. Source: The figure is from Reference 18.
Figure 16.7 shows the numerical solution of Equation 16.1 for |E| and θ versus x at z = 400. The other parameters are ν = 50, am = 0.2, and k = 0. A classical DSW, as seen in Figure 16.6, has a structure where the waves decrease in amplitude from the front to the rear. In this nonlocal case, this pattern disappears with the waves no longer ordered by amplitude. Also, it can be seen from the figure that there are less director peaks in the bore than electric field peaks. The larger nonlocality of this example has resulted in a broad director response where the director pulses are wider than the electric field pulses (β = 6.9 > w = 2.9). This generates a wide potential well enclosing all the solitary waves in the DSW, which wipes out many individual peaks. The solitary waves interact with each other in a nonlocal manner through this potential well, and the amplitude of the highest solitary wave varies in a complicated manner with z, because of this nonlocal interaction. However, the average maximum amplitude, a = 0.55 and α = 0.11, are very close to the estimates from uniform soliton theory; a = 0.53, and α = 0.11.
Figure 16.7 Numerical solutions of Equation 16.1 versus x at z = 400. Shown are |E| (solid line) and θ (dashed line). The other parameters are ν = 50, am = 0.2, and k = 0. Source: The figure is from Reference 18.
16.6 Conclusions
Experimental, numerical, and theoretical results have been reviewed for the development of DSW in nematic liquid crystal and a range of other optical media. A key focus has been the presentation of a method, uniform soliton theory, to obtain semianalytical results for DSW in a nonlocal focusing medium. Focusing media are subject to MI but a range of emerging experimental results show the establishment of a DSW, before MI destroys the bore structure.
The semianalytical solutions, which predict the amplitude of the largest wave in the DSW, were developed using uniform soliton theory that uses conservation laws, and the assumption that a train of uniform solitary waves is generated by the shock. The semianalytical predictions, for solitary wave amplitude, are found to be in excellent agreement with numerical solutions for (1 + 1)D bores.
This represents a novel method of finding analytical solutions for a focusing medium and will motivate additional experimental observations of DSW in a range of focusing and defocusing media. A key aim of future experimental studies should be measurement of the amplitude of solitary waves generated in a DSW, which will allow validation of the semianalytical theory. The theory can also be applied to other NLS-type equations of focusing type. These include NLS-type equations with saturable nonlinearity, a nonlocal version with a Gaussian or other nonexponential response function, and also the implicit nonlinearity that occurs for applications involving colloidal media.
Moreover, experiments are usually performed for DSW with both one- and two-dimensional geometries. The results presented here describe one-dimensional DSW but not two-dimensional ones [18] extended their one-dimensional results to obtain approximate results for two-dimensional DSW with circular symmetry, but the results were only valid for DSW with large radius. Future work could extend uniform soliton theory to two-dimensional DSW of arbitrary radius. This would involve using the variational approach to find an approximate annular solitary wave solution, which would have the appropriate symmetry for two-dimensional DSW.
There has been much experimental work on DSW in defocusing nonlocal media. Uniform soliton theory should also be a useful tool in predicting the amplitude of dark solitary waves, in the DSW generated in these media. The local defocusing NLS equation shows that the DSW is usually part of a more complicated structure involving rarefaction waves, vacuum points, and plateau regions. To accurately predict the amplitude of dark solitary waves in defocusing nonlocal media, all of the solution types need to be analytically modeled; a much more challenging task than in the case of nonlocal focusing equations.
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