# Dispersive shock waves

Post-publication activity

Curator: Mark Ablowitz

Shock waves can be generated by an initial abrupt change in a physical quantity and involve flow speeds larger than the local speed of sound. A shock wave propagating through a medium, such as a fluid, exhibits a rapid change in the material properties that characterize the medium. In viscous media, i.e. for a viscous shock wave (VSW), the change is typified by a nearly discontinuous jump from one value to another. On the other hand, in dispersive media the change in values via a dispersive shock wave (DSW) typically occurs via a modulated wavetrain. A one-dimensional example of a dispersive shock wave in the Korteweg-deVries equation, is shown in Figure 1.

Figure 1: Dispersive shock wave in the Korteweg-deVries equation.

The dynamical behavior of a medium such as a fluid can often be described in terms of a set of conservation laws modified by a small amount of dissipation and/or dispersion. When dissipation and dispersion are neglected, a large class of initial data can lead to a derivative discontinuity or a gradient catastrophe whereby the solution develops an infinite derivative in finite time. This singular behavior is unphysical and therefore the effects of small dissipation and/or small dispersion require inclusion in the model. When dissipation dominates dispersion, as is often the case in classical fluids such as a compressible gas, a dissipative regularization procedure can be employed which leads to generalized (weak) solutions that can include discontinuities which represent viscous shock waves. If finite but small viscosity is included in the model, then the solutions are smooth but change rapidly from one value to another. However, when dispersion dominates dissipation, for example in the quantum superfluidic Bose-Einstein condensate, a dispersive regularization procedure is needed. The solution for small dispersion (which dominates dissipation) involves nonlinear wave averaging and has a modulated wavetrain that allows the transition from one value of the medium to another.

In the physical and applied mathematics literature, dispersive shock waves have been referred to in various ways: collisionless shock wave, dissipationless shock wave, non-dissipative shock wave, and undular bore. Note that an undular bore in shallow water displays dispersive shock behavior but a turbulent bore, for example a tidal bore, is typically characterized by dissipative effects, and is usually described as a viscous shock wave.

# Background

Figure 2: Experimental absorption images of the Bose-Einstein condensate blast wave problem performed at JILA, University of Colorado, Boulder (Hoefer et al. 2006). The oscillatory ring structures correspond to dispersive shock waves.
Figure 3: Experimental images of two one-dimensional (left) and two interacting two-dimensional (right) optical DSWs performed at Princeton University (Wan, Jia, Fleischer 2007).

One of the earliest observations of dispersive shock waves, though not described as such at the time, occurred in the context of water waves where, using current nomenclature, small amplitude bores were studied (Bazin 1850). Since then, there have been numerous observations of dispersive shock behavior. In laboratory experiments in 1970, collisionless ion-acoustic shock waves were observed from the interaction of two plasmas (Taylor, Baker, Ikezi 1970). Optical wave breaking was observed in the propagation of light through a nonlinear fiber in 1989 (Rothenberg, Grischkowsky 1989). More recently, dispersive shock waves were created in a Bose-Einstein condensate by creating a density notch with a slow light technique (Dutton et al. 2001). Two-dimensional dispersive shock waves emanating from a laser blast in a Bose-Einstein condensate were later observed (Hoefer et al. 2006); see Figure 2--see also (Simula et al. 2005) where shock waves associated with a rotating Bose-Einstein condensate were observed. The propagation of an intense electromagnetic wave through a photorefractive medium excited dispersive shock waves and interactions in one and two dimensions (Wan, Jia, Fleischer 2007); see Figure 3. Experimental studies of these nonlinear, dispersive media continue (Barsi et al. 2007), (Jia, Wan, Fleischer 2007), (Ghofraniha et al. 2007), (Chang, Engels, Hoefer 2008), (Hoefer, Engels, Chang 2008), (Conti et al. 2009), (Barsi, Wan, Fleischer 2009).

## Regularization Via the Method of Averaging

Dispersive shock waves typically involve two scales: a fast, oscillatory wave scale and a slow, modulation scale. The DSW wavetrain when viewed locally, i.e. over a small region of space and time, can display periodic structure. However, over a larger region covering multiple wave oscillations it can reveal slow modulation of the wave parameters. Thus a slowly modulated wavetrain involves a fast scale--the phase of the wave--and a slow scale--the modulation of the wave's parameters such as the amplitude, wavelength and frequency of the wave (e.g. see Figure 1). As such, asymptotics and multiple scale analysis are useful tools to study their behavior.

In 1965, Whitham formulated an asymptotic averaging method for a class of nonlinear partial differential equations (PDEs) which have special (single phase) periodic traveling wave solutions. Whitham's original method can be viewed as a generalization of the method of averaging for ordinary differential equations (Whitham 1965). This results in the so-called Whitham modulation equations which describe the slow modulation of a rapidly oscillating wave. Indeed, these equations were used in (Gurevich, Pitaevskii 1974) to originally construct an asymptotic representation of a dispersive shock wave in the Korteweg-de Vries (KdV) equation. The KdV equation arises in many physical situations; e.g. water waves, plasma physics, and lattice dynamics (cf. Ablowitz, Segur 1981).

Once the Whitham modulation equations have been constructed, it is necessary to impose appropriate initial/boundary data for their solution. This procedure will be demonstrated by two examples below. Note that some progress has been made in understanding the behavior of the dispersive Riemann problem without requiring the full solution, and hence diagonalization, of the Whitham equations (El 2005).

## Averaging and the Korteweg-de Vries Equation

Whitham averaged over the first three KdV conservation laws. This yielded modulation equations for the slowly varying parameters of the underlying periodic solution. Whitham was able to diagonalize, through non-trivial algebraic manipulations, these modulation equations. This resulted in a system of three quasi-linear first order hyperbolic PDEs in time and space. But, in general, quasi-linear hyperbolic PDEs can only be diagonalized for systems of two equations (cf. Whitham 1974). Whitham's ability to diagonalize the KdV modulation equations and therefore solve them in principle, is closely related to the fact that the Korteweg-de Vries Equation is completely integrable.

Whitham's method can be extended to multiple phase wavetrains (Ablowitz, Benney 1970). By use of inverse spectral theory, finite gap integration, and algebraic geometry, it was shown in (Flaschka, Forest, McLaughlin 1980) that the extension of Whitham theory to multi-phase averaging can also be viewed as a modulation of an associated Riemann surface. The key unknowns associated with the Whitham modulation equations, the so-called Riemann invariants, were shown to be the branch points of the associated Riemann surface. The benefits of this approach include the direct diagonalization of the Whitham modulation equations. Since then, a number of integrable equations and their associated Whitham modulation equations have been studied using this method, including and especially the Nonlinear Schrodinger equation (Forest, Lee 1986), (Pavlov 1987), (Levermore 1988), (Kamchatnov 2000).

# Dispersive Riemann Problem

The Riemann problem consists of two initial, constant states separated by a discontinuity. This problem was introduced in the context of classical gas dynamics by Riemann in 1860. Since gas dynamics is intrinsically dissipative, the Riemann problem has traditionally been associated with a dissipative regularization leading to viscous shock waves. The Riemann problem takes on new meaning when viewed in the context of a dispersive regularization (Gurevich, Pitaevskii 1974), leading to dispersive shock waves.

## Example: Korteweg-de Vries Equation

The dispersive Riemann problem for the Korteweg-de Vries equation is written $\tag{1} u_t + \left( \frac{1}{2} u^2 + \varepsilon^2 u_{xx} \right)_x = 0,$

$\tag{2} u(x,0) = \left \{ \begin{array}{cc} u_l & x < 0 \\ u_r & x > 0 \end{array} \right ., \quad \varepsilon^2 \ll 1 .$

This initial value problem was asymptotically solved in (Gurevich, Pitaevskii 1974). An outline of the construction of their asymptotic solution is presented here.

If $$u_l < u_r\ ,$$ then a rarefaction wave solution to the Hopf equation $u_t + u u_x = 0,$ describes the leading order behavior. The solution is $u(x,t) \sim \left \{ \begin{array}{cc} u_l & x < u_l t \\ x/t & u_l t < x < u_r t \\ u_r & u_r t < x \end{array} \right . .$ The case $$u_l > u_r$$ gives rise to a dispersive shock wave and requires a more careful asymptotic analysis which is now explained.

The method of Whitham averaging requires a periodic, traveling wave solution to eq. (1) in the form $u(x,t) = \phi(\xi), \quad \xi = \frac{x-Vt}{\varepsilon},$ where $$V$$ is the wave's phase speed. Inserting this ansatz into eq. (1) results in $-V \phi' + \phi \phi' + \phi''' = 0 .$ Integrating this ordinary differential equation twice leads to $\tag{3} (\phi')^2 = -\frac{1}{3}(\phi^3 - 3V \phi^2 + A \phi + B) \equiv \frac{1}{3} P(\phi),$

where $$A$$ and $$B$$ are arbitrary integration constants. Solutions to equations of this form, when the polynomial $$P$$ is a cubic or quartic polynomial, are elliptic functions. The cubic polynomial $$P$$ can be written in terms of its roots $\tag{4} P(\phi) = (\lambda_1 - \phi)(\lambda_2 - \phi)(\lambda_3 - \phi), \quad \lambda_1 \le \lambda_2 \le \lambda_3 .$

It is convenient to make the following linear transformation $r_1 = \frac{1}{2}(\lambda_1 + \lambda_2), \quad r_2 = \frac{1}{2}(\lambda_1 + \lambda_3), \quad r_3 = \frac{1}{2}(\lambda_2+\lambda_3), \quad r_1 \le r_2 \le r_3.$ Then equation (1) admits the following family of traveling wave solutions to eq. (3) $\tag{5} u(x,t) = \phi(\xi;\vec{r}), \quad \xi = \frac{x-Vt}{\varepsilon},$

$\phi(\xi; \vec{r}) = r_1 + r_2 - r_3 + 2(r_3 - r_1) \textrm{dn}^2\left( \sqrt{\frac{r_3 - r_1}{6}} \xi; m \right),$ $m = \frac{r_2 - r_1}{r_3 - r_1}, \quad V = \frac{1}{3}(r_1 + r_2 + r_3),$ $\vec{r} = [r_1,r_2,r_3], \quad r_1 \le r_2 \le r_3.$ where dn is a Jacobi elliptic function. This periodic solution depends on the three parameters $$r_1, r_2, r_3\ ,$$ or, by eq. (3), the roots $$\lambda_1, \lambda_2, \lambda_3$$ of the cubic polynomial $$P$$ in (4). The wavelength of oscillation in the variable $$\xi$$ is $\tag{6} L = 2K[m] \sqrt{\frac{6}{r_3 - r_1}},$

where $$K[m]$$ is the complete elliptic integral of the first kind. The amplitude of wave oscillation is $\tag{7} A = 2(r_2 - r_1) .$

The KdV equation, being completely integrable, admits an infinite number of associated conservation laws. Because there are three modulation parameters, averaging of three of these conservation laws is required. A uniqueness result was proved in (Flaschka, Forest, McLaughlin 1980) where it was shown that any three of the infinite number of conservation laws for the KdV equation can be averaged and the same Whitham modulation equations will result.

Whitham's asymptotic, nonlinear wave averaging method proceeds as follows (Whitham 1965). There are two natural scales associated with Eq. (1), the fast oscillatory scale $$\xi = (x-Vt)/\varepsilon = O(1/\varepsilon)$$ and the slow modulation scale $$|x|, |t| \ll 1/\varepsilon\ .$$ Due to the choice of initial conditions (2), it is reasonable to assume a-priori that the solution is asymptotically described by the rapidly oscillating wave $$\phi$$ of Eq. (5) with slow variation of its parameters $$\vec{r} = \vec{r}(x,t)\ .$$ It is later shown that this ansatz can be chosen in such a way as to agree with the initial data in an appropriate sense. As a consequence of this assumption, it is necessary to fix the period of oscillation to a constant value in order to prevent secular growth. For example, assuming the nonconstant period $$L$$ in (6), then $\frac{\partial}{\partial x} \phi(\xi;\vec{r}) = \frac{\partial}{\partial x} \phi(\xi + nL;\vec{r}) = (n \frac{\partial L}{\partial x} + \xi_x) \phi'(\xi;\vec{r}) + \sum_{i = 1}^3 \frac{\partial \phi}{\partial r_i} \frac{\partial r_i}{\partial x},$ for any integer $$n$$ and can be made arbitrarily large. The constant period $$2\pi$$--one can choose any constant--is introduced through $\tag{8} \tilde{\phi}(\theta;\vec{r}) = \phi\left(\frac{K[m]\sqrt{6}}{\pi\sqrt{r_3 - r_2}} \theta; \vec{r} \right) = r_1 + r_2 - r_3 + 2(r_3 - r_1) \textrm{dn}^2\left( \frac{K[m]}{\pi} \theta; m \right),$

where the new fast phase $$\theta$$ satisfies the generalized wavenumber and frequency relations $\tag{9} \theta_x = \frac{2\pi}{L \varepsilon} = \frac{k}{\varepsilon} = \frac{\pi}{\varepsilon K[m]} \sqrt{\frac{r_3 - r_1}{6}},$

$\theta_t = - \frac{\omega}{\varepsilon} = - \frac{k V}{\varepsilon} = -\frac{1}{3}(r_1 + r_2 + r_3) \frac{\pi}{\varepsilon K[m]}\sqrt{\frac{r_3 - r_1}{6}} ,$ standard for modulated nonlinear waves (Whitham 1974). As a result of compatibility, one finds the conservation of waves $\theta_{xt} = k_t/\varepsilon = \theta_{tx} = -\omega_x/\varepsilon ~ \Rightarrow ~ k_t + \omega_x = 0 .$

As an example of the averaging procedure, the first KdV conservation law will be considered--that is the equation itself $u_t + \left( \frac{1}{2} u^2 + \varepsilon^2 u_{xx} \right)_x = 0.$ Inserting the solution from Eq. (8) and averaging over the fast oscillations according to $\overline{F(\tilde{\phi})} = \frac{1}{2\pi} \int_0^{2\pi} F(\tilde{\phi}(\theta;\vec{r})) d\theta,$ while assuming slow variation in the parameters results in the leading order equation $\left( \overline{\tilde{\phi}} \right)_t + \left( \frac{1}{2}\overline{\tilde{\phi}^2} + \overline{\tilde{\phi}''} \right)_x = 0 .$ Due to the fixed, constant period, the averaging operation and the partial derivatives $$\partial/\partial t\ ,$$ $$\partial/\partial x$$ commute at leading order. The required averages can be computed directly in terms of elliptic integrals $\tag{10} \begin{array}{rcl} \overline{\tilde{\phi}} &=& r_1 + r_2 - r_3 + 2(r_3 - r_1)\frac{E[m]}{K[m]}, \\ \overline{\tilde{\phi}^2} &=& (r_1+r_2-r_3)^2 - \frac{4}{3}(r_3-r_2)(r_3-r_1) + \frac{4}{3}(r_3-r_1)(r_1+r_2+r_3) \frac{E[m]}{K[m]}, \\ \overline{\tilde{\phi}''} &=& 0 , \end{array}$

where $$E[m]$$ is the complete elliptic integral of the second kind. This result combined with two other averaged conservation laws gives the Whitham modulation equations for the slowly varying parameters $$r_1, r_2, r_3\ .$$ With considerable manipulation (Whitham 1965) (see also (Kamchatnov 2000) where Whitham's calculation is performed in detail) or the finite gap integration method (Flaschka, Forest, McLaughlin 1980), one can obtain the KdV modulation equations in diagonal Riemann invariant form $\tag{11} \frac{\partial r_i}{\partial t} + v_i(\vec{r}) \frac{\partial r_i}{\partial x} = 0, \quad v_i = V - \frac{L}{3\frac{\partial L}{\partial r_i}} .$

The velocities are explicitly given in terms of elliptic integrals. For example, $\tag{12} v_2 = \frac{1}{3}(r_1 + r_2 + r_3) - \frac{2}{3}(r_2 - r_1)\frac{(1-m)K[m]}{E[m]-(1-m)K[m]} .$

It has been proven that the KdV Whitham equations are strictly hyperbolic (e.g. $$v_1 < v_2 < v_3$$) and genuinely nonlinear (e.g. $$\partial v_i/\partial r_i \ne 0$$) (Levermore 1988).

Now that the Whitham equations have been constructed, it is necessary to prescribe appropriate initial and/or boundary data for their solution. For the initial data in eq. (2), there are two equivalent methods to do this which are termed here as global and matched regularization, respectively. The global regularization method will be presented in detail below.

### Matched Regularization

The matched regularization procedure, introduced in (Gurevich, Pitaevskii 1974)--see also (El 2005)--involves a moving boundary value problem. Until a point of gradient catastrophe $$(x,t) = (x_b,t_b)\ ,$$ the solution is, to leading order, described by the Hopf equation $\tag{13} {u_0}_t + u_0 {u_0}_x = 0 .$

For $$t > t_b\ ,$$ a modulated wave region, the DSW, opens up with left and right boundaries $$x_-(t)$$ and $$x_+(t)\ ,$$ respectively. The interior of the modulation region is described by Eq. (8) with the parameters $$\vec{r}$$ satisfying the Whitham equations (11). The boundary of the modulated region is matched to the smooth, dispersionless flow in an average sense $\overline{\tilde{\phi}}(x_\pm(t),t) = u_0(x_\pm(t),t) .$ The left and right boundaries are double characteristics where two Riemann invariants coincide. For the left boundary, $r_1(x_-(t),t) = r_2(x_-(t),t),$ so that, according to Eq. (10) $\overline{\tilde{\phi}}(x_-(t),t) = r_3(x_-(t),t) = u_0(x_-(t),t),$ and $\frac{d x_-}{dt} = \lim_{r_2 \to r_1} v_2(r_1,r_2,r_3), \quad x_-(t_b) = x_b .$ The right boundary is determined from $r_2(x_+(t),t) = r_3(x_+(t),t),$ so that $\overline{\tilde{\phi}}(x_+(t),t) = r_1(x_+(t),t) = u_0(x_+(t),t),$ and satisfies $\frac{d x_+}{dt} = \lim_{r_2 \to r_3} v_2(r_1,r_2,r_3), \quad x_+(t_b) = x_b .$ A necessary condition is $$x_-(t) \le x_+(t)$$ which has been identified with the entropy condition for shock waves (El 2005).

### Global Regularization

The global regularization procedure, introduced in (Bloch, Kodama 1992)--see also (Kodama 1999)-- is implemented by choosing initial data for the Whitham equations, $$\vec{r}(x,0)\ ,$$ that characterize the initial data in eq. (2) in an average sense $\overline{\tilde{\phi}}(x,0) = \left \{ \begin{array}{cc} u_l & x < 0 \\ u_r & x > 0 \end{array} \right . ,$ and lead to a global solution of the Whitham equations. Both properties can be achieved by the following choice of initial data $\tag{14} r_1(x,0) = u_r, \quad r_2(x,0) = \left \{ \begin{array}{cc} u_r & x < 0 \\ u_l & x > 0 \end{array} \right., \quad r_3(x,0) = u_l .$

The points where $$r_1 = r_2$$ or $$r_2 = r_3$$ are to be understood in a limiting sense, e.g. $$r_2 \to r_3^-\ ,$$ when calculating the average $$\overline{\tilde{\phi}}(x,0)\ .$$

Figure 4: Global solution to the KdV Whitham equations.

Using the above global regularization procedure, the global solution to the KdV Whitham equations with initial data in eq. (14) is a self-similar rarefaction wave solution (depending only on $$x/t$$) satisfying $r_1(x,t) = u_r, \quad r_2(x,t) = \left \{ \begin{array}{cc} u_r & x/t < v_- \\ r(x/t) & v_- < x/t < v_+ \\ u_l & v_+ < x/t \end{array} \right ., \quad r_3(x,t) = u_l,$ where $$r(x/t)$$ satisfies the implicit relation $v_2(u_r,r(x/t),u_l) = x/t .$ For a plot of the solution, see Figure 4. The trailing and leading edge front speeds are determined by studying the limiting behaviors $$r \to u_r^+$$ and $$r \to u_l^-$$ of Eq. (12), respectively $v_- = \lim_{r \to u_r^+} v_2(u_r,r,u_l) = -u_l + 2u_r,$ $\quad v_+ = \lim_{r \to u_l^-} v_2(u_r,r,u_l) = \frac{1}{3}(2u_l + u_r) .$

This solution to the modulation equations enables the construction of a KdV dispersive shock wave whose explicit, asymptotic representation is $\tag{15} u(x,t) \underset{\varepsilon \ll 1}{\sim} r(x/t) - \Delta + 2 \Delta \textrm{dn}^2 \left ( \frac{K[m(x/t)]}{\pi}\theta(x,t);m(x/t) \right ),$

$\Delta = u_l - u_r, \quad m = \frac{r(x/t)-u_r}{\Delta},$ where the phase $$\theta$$ satisfies the following generalized wavenumber and frequency relations from Eq. (9) $\tag{16} \theta_x = \frac{k}{\varepsilon} = \frac{\pi}{\varepsilon K[m]} \sqrt{\frac{\Delta}{6}}, \quad \theta_t = - \frac{k V}{\varepsilon} = -\frac{1}{3}(u_r + r(x/t) + u_l) \frac{\pi}{\varepsilon K[m]}\sqrt{\frac{\Delta}{6}} .$

These relations can be integrated in the oscillatory region to give $\tag{17} \theta(x,t) = - \frac{\pi}{\varepsilon} \sqrt{\frac{\Delta}{6}} \int_x^{v_+ t} \frac{dx'}{K[m(x'/t)]} + \theta_0(x,t), \quad v_-t \le x \le v_+ t .$

The slow variation of the phase $$\theta_0(x,t)$$ is, in general, not determined by leading-order Whitham theory. However, it is natural to take $$\theta_0(x,t) \equiv 0\ ,$$ as will be shown shortly.

There are two associated shock speeds $$v_\pm\ ,$$ that of the leading and trailing edges already derived from the solution to the Whitham modulation equations. These speeds can also be understood by studying the asymptotic KdV DSW from eq. (15) in two limiting regimes: small wavenumber $$k \to 0$$ and small amplitude $$A \to 0\ .$$

As $$k \to 0$$ in eq. (16), $$K[m] \to \infty$$ so that $$m \to 1$$ or $$r \to u_l^-\ .$$ This long wavelength limit and the behavior of the Jacobi elliptic function $$\textrm{dn}$$ as $$m \to 1$$ suggests that one of the DSW fronts can be viewed as a KdV soliton propagating on the background $$u_r\ .$$ Its phase speed from eq. (16) is $\lim_{r \to u_l^-} V = \lim_{r \to u_l^-} \frac{1}{3}(u_r + r + u_l) = \frac{1}{3}(2u_l + u_r) = v_+,$ and therefore is identified with the leading edge of the DSW. The soliton amplitude from eq. (7) is $\lim_{r \to u_l^-} A = \lim_{r \to u_l^-} 2(u_l - u_r) = 2 \Delta ,$ so that the soliton has the form $u(x,t) \sim u_r + 2\Delta \textrm{sech}^2 \left( \sqrt{\frac{\Delta}{6\varepsilon^2}} (x-v_+ t) \right) .$ The slow variation of the phase $$\theta_0(x,t) \equiv 0$$ in Eq. (17) is fixed by requiring that the soliton is centered at the leading edge. A plot of this solution is shown in Figure 5. In any physical system modeled by the KdV equation, there will be a finite dispersion coefficient $$\varepsilon > 0\ .$$ Mathematically it is natural to consider the limit $$\varepsilon \to 0\ .$$ The appropriate limit is a weak limit and is the average $$\overline{u}$$ (Lax, Levermore 1983), (Venakides 1985), also plotted in Figure 5.

Figure 5: KdV DSW at $$t = 1$$ (solid), its envelope (dash-dotted), and its average (dashed). The points marked correspond to phase portraits in Figure 5.

The trailing edge can be viewed as a packet of small amplitude waves, $$A \to 0$$ or, from eq. (7), $$r \to u_r^+\ ,$$ propagating on the background $$u_l$$ with a certain group velocity. The group velocity (Whitham 1974) for KdV linear waves with wavenumber $$k$$ propagating on the background $$u_l$$ is $v_g = u_l - 3k^2 .$ The appropriate wavenumber can be determined by evaluating $$k$$ from eq. (16) $\lim_{r \to u_r^+} k = \lim_{r \to u_r^+} \frac{\pi}{K[m]}\sqrt{\frac{\Delta}{6}} = \sqrt{\frac{2\Delta}{3}} ,$ because $$K[0] = \pi/2\ .$$ Therefore, the trailing edge of the DSW consists of a small amplitude wave packet propagating with the group velocity $v_g = u_l - 2 \Delta = -u_l + 2u_r = v_- .$ A KdV DSW can be interpreted as a nonlinear wave packet with wavenumbers lying in the range $$k \in [0,\sqrt{2\Delta/3}]$$ and amplitudes lying in the range $$A \in [0,2\Delta]\ .$$

Using inverse scattering theory, Khruslov in (Khruslov 1975) and (Khruslov 1976) solved the initial step problem rigorously, demonstrating that an increasing number of asymptotic solitons appear in an expanding neighborhood of the leading edge. While not exact solutions to the KdV equation, these structures asymptotically approach true soliton solutions as $$\varepsilon$$ decreases.

Figure 6: Example phase portraits of the KdV DSW. Each phase portrait labeled $$x_i$$ corresponds to a point in the DSW solution of Figure 5 with parameters $$\vec{r}$$ and a specific periodic solution $$\tilde{\phi}$$ from Eq. (8). As the DSW is traversed from trailing to leading edge, the hashed region is filled by all the phase portraits.

It is also possible to interpret the KdV DSW from a dynamical systems perspective. Five points on the DSW shown in Figure 5 $$x_i\ ,$$ $$i=0,1,2,3,4$$ correspond to five different phase portraits in Figure 6. Due to the two-scale nature of the DSW, in the vicinity of each point $$x_i\ ,$$ the DSW in Eq. (15) is approximately the exact, periodic solution $$\tilde{\phi}$$ in Eq. (5) with the parameters $$\vec{r}$$ determined by the solution to the Whitham equations. Once the solution to the Whitham equations is determined, a phase space corresponding to this family of periodic orbits can be associated with the DSW and visualized via phase portraits shown in Figure 6. The solution at the points $$x_0$$ and $$x_4$$ in Figure 5 correspond to fixed points in Figure 6. The points $$x_1$$ and $$x_2$$ correspond to periodic orbits and the point $$x_3$$ corresponds to a homoclinic orbit, the KdV soliton. Since these phase portraits progressively expand as one moves through the DSW from small amplitude oscillations at the trailing edge to the soliton at the leading edge, the phase space associated with a DSW encompasses all the solution curves enclosed by the soliton homoclinic orbit.

### General Initial Value Problem

The dispersive Riemann problem considered in Eq. (1) is a particular example that enables an explicit and direct study of a single DSW for the KdV equation and provides physically relevant information about the DSW profile and speeds. The general initial value problem $u_t + \left( \frac{1}{2} u^2 + \varepsilon^2 u_{xx} \right)_x = 0,$ $u(x,0) = u_0(x), \quad \varepsilon^2 \ll 1,$ where $$u_0(x)$$ is restricted to some class of functions, e.g. smooth and rapidly decaying, has been studied in many works. If the initial data is decreasing $$(u_0'(x) < 0)$$ anywhere, then there is a critical point $$(x,t) = (x_b,t_b)\ ,$$ the breaking point, where the solution will develop a region of rapid oscillations with wavelength of order $$\mathcal{O}(\varepsilon)\ .$$ Beyond the breaking point, the oscillatory region was rigorously shown to be described by the Whitham modulation equations in the limit $$\varepsilon \to 0$$ (Lax, Levermore 1983), (Venakides 1985). The limiting solution is to be understood in a weak (averaged) sense.

A careful numerical study of the oscillatory region was undertaken in (Grava, Klein 2007) where it was shown that different portions converge to the Whitham modulation solution at different asymptotic rates in $$\varepsilon\ .$$ A rigorous asymptotic study in (Claeys, Grava 2009) for $$(x,t)$$ close to $$(x_b,t_b)$$ demonstrated that the solution is described by the fourth order ODE $\tag{18} X = T U - \left [ \frac{1}{6} U^3 + \frac{1}{24}(U_X^2 + 2 U U_{XX}) + \frac{1}{240} U_{XXXX} \right ],$

where $$X, T, U$$ are rescalings of $$x, t, u\ .$$ Further work in (Grava, Klein 2008) and (Claeys, Grava 2009) showed that the leading, soliton edge is asymptotically described by a certain solution to $q''(s) = s q + 2 q^3(s) .$

The rigorous studies just mentioned rely on the integrability of the KdV equation. Whitham averaging theory via the matched regularization procedure discussed earlier is not tied to integrability of the governing equation and can be used to construct an asymptotic solution that is valid in some situations. It should be noted that by assuming that a dispersive shock wave has vanishing amplitude waves on one end and large wavelength (vanishing wavenumber), finite amplitude waves at the other, it has been shown that the leading and trailing edge behavior can be analyzed for a class of dispersive PDE without integrating the full Whitham equations (El 2005). However, leading order Whitham theory does not determine the slow variation of the phase $$\theta_0(x,t)$$ in Eq. (17) and a generalization to multiple rapidly oscillating phases is required if the Whitham equations themselves develop a gradient catastrophe (Flaschka, Forest, McLaughlin 1980), (Grava, Tian 2002).

## Example: Nonlinear Schrodinger Equation

An instructive example of a dispersive regularization for a system of conservation laws involves the Nonlinear Schrodinger equation (NLS) $\tag{19} i \varepsilon \psi_t = -\frac{\varepsilon^2}{2} \psi_{xx} + |\psi|^2 \psi ,$

where $$0 < \varepsilon \ll 1$$ is proportional to the wavelength and period of oscillation in the system. This equation models the slowly varying envelope of an electromagnetic wave in a nonlinear, defocusing medium. It also models the dynamics of a Bose-Einstein condensate. In both systems, dispersive shock waves have been observed experimentally (Wan, Jia, Fleischer 2007), (Simula et al 2005), (Hoefer et al 2006) (see Figure 2 and Figure 3).

The NLS equation admits a representation in terms of fluid-like variables through the transformation $\psi = \sqrt{\rho}e^{\frac{i}{\varepsilon} \phi}, \quad u = \phi_x .$ Inserting this ansatz into eq. (19) and equating real and imaginary parts leads to $\tag{20} \rho_t + (\rho u)_x = 0, \quad (\rho u)_t + \left ( \rho u^2 + \frac{1}{2}\rho^2 \right )_x = \frac{\varepsilon^2}{4} \left ( \rho (\log\rho)_{xx} \right )_x .$

When $$\varepsilon = 0\ ,$$ these equations are equivalent to the Euler equations of gas dynamics with pressure law $$p = \frac{1}{2}\rho^2$$ or also the shallow water equations and can be diagonalized in the form $\tag{21} \frac{\partial r_\pm}{\partial t} + v_\pm \frac{\partial r_\pm}{\partial x} = 0, \quad r_\pm = u \pm 2\sqrt{\rho}, \quad v_\pm = \frac{1}{4}(3r_\pm + r_\mp) .$

Without loss of generality, the dispersive Riemann problem for eqs. (20) is parametrized in terms of two parameters $$\rho_0 > 1$$ and $$u_0 \in \mathbb{R}$$ $\tag{22} \rho(x,0) = \left \{ \begin{array}{cc} \rho_0 & x < 0 \\ 1 & x > 0 \end{array} \right . , \quad u(x,0) = \left \{ \begin{array}{cc} u_0 & x < 0 \\ 0 & x > 0 \end{array} \right . .$

This problem was solved in (Gurevich, Krylov 1987) and (El et al. 1995).

Similar to the KdV case, there exists a periodic, traveling wave solution to eqs. (20) with phase speed $V = \frac{1}{4}(r_1+r_2+r_3+r_4),$ and wavelength $L/\varepsilon = \frac{4 K[m]}{\varepsilon \sqrt{(r_4-r_2)(r_3-r_1)}} .$ The Whitham modulation equations are $\frac{\partial r_i}{\partial t} + v_i(\vec{r}) \frac{\partial r_i}{\partial x} = 0, \quad v_i(\vec{r}) = V - \frac{L}{4 \frac{\partial L}{\partial r_i}} , \quad i = 1, 2, 3, 4,$ except this time, for the NLS eq., there are four parameters rather than three. The dispersive regularization procedure proceeds in a manner similar to the KdV case discussed earlier. The result depends on the two parameters $$\rho_0$$ and $$u_0\ .$$ A classification diagram is shown in Figure 7 and asymptotic solutions from six different points are shown in Figure 7.

Figure 7: Classification of solutions to the NLS dispersive Riemann problem. RW = rarefaction wave and DSW = dispersive shock wave. The symbols $$\circ\ ,$$ $$\ddagger\ ,$$ $$\triangle\ ,$$ $$\ast\ ,$$ $$\times\ ,$$ and $$+$$ correspond to values $$(\rho_0,u_0)$$ with corresponding asymptotic solutions shown in Figure 7.
Figure 8: Example asymptotic solutions to the NLS dispersive Riemann problem (solid) along with the average $$\overline{\rho}$$ (dashed). The symbols $$\circ\ ,$$ $$\ddagger\ ,$$ $$\triangle\ ,$$ $$\ast\ ,$$ $$\times\ ,$$ and $$+$$ in the upper right corners correspond to choices of $$(\rho_0,u_0)$$ in the phase diagram of Figure 7.

In general, the solution consists of two waves which are rarefaction waves (RWs) or dispersive shock waves (DSWs). For example, the case $$\ast$$ in Figure 7 and Figure 8 corresponds to the so-called shock tube problem where there is an initial jump in density but zero initial fluid velocity. The leading order solution is a rarefaction wave propagating to the left and a dispersive shock wave propagating to the right. Both types of waves are called simple waves because only one Riemann invariant changes across the wave. In the case of a rarefaction wave, one of $$r_\pm$$ changes while the other is constant. For a dispersive shock wave, one of the $$r_i\ ,$$ $$i = 1, ~ 2, ~ 3,$$ or $$4$$ varies while the rest are constant. It can be shown by a dispersive regularization that the behavior of the solution depends on the relative ordering of the four values $u_0 \pm 2\sqrt{\rho_0}, \quad \pm 2 ,$ which are precisely the values of the Euler Riemann invariants $$r_\pm$$ in eq. (21) evaluated at the initial data in eq. (22). There are five possible orderings of these values leading to five qualitatively different solution behaviors which are shown in Figure 7 and Figure 8. For sufficiently small initial velocities $$u_0 \le -2\sqrt{\rho_0}+2$$ (cases $$\times$$ and $$+$$ in Figure 7, Figure 8), there are no shock waves in the solution, only rarefaction waves. For sufficiently large velocities $$u_0 > 2\sqrt{\rho_0} + 2$$ (cases $$\circ$$ and $$\ddagger$$ in Figure 7, Figure 8), the asymptotic solution consists of two dispersive shock waves connected by a periodic wavetrain.

It is possible for a dispersive shock wave to develop a point where the density $$\rho$$ is zero, i.e. a vacuum point (El et al. 1995). When this occurs, the fluid flows into the DSW from both sides. It can be shown that a vacuum point occurs in the rightmost dispersive shock wave arising in the solution of the dispersive Riemann problem when $$u_0 \ge -2\sqrt{\rho_0}+6$$ (see $$\ddagger\ ,$$ $$\triangle$$ in Figure 7 and Figure 8). A problem that is dual to the shock tube is the case when the initial density is constant $$\rho_0 = 1$$ and there is only a jump in the velocity. In this case, when $$u_0 > 4\ ,$$ a standing wave with a sequence of vacuum points forms that connects two dispersive shock waves (see $$\circ$$ in Figure 8). An initial jump in the velocity corresponds to the interaction of two uniform plane waves in the NLS equation. This problem is the nonlinear analogue of quantum interference when two plane waves of the linear Schrödinger equation are superposed. In this nonlinear case, the standing wave is an elliptic function whereas in the linear case, the standing wave is a trigonometric function. In the limit of small amplitude, the two results coincide.

# Related Problems

There is a growing literature of the theory of dispersive shock waves and applications that is too numerous to list exhaustively. Some results related to the KdV and NLS equations are listed below.

The initial value problem for the KdV equation in the small dispersion limit has engendered a large number of works, the first of which include (Lax, Levermore 1983) and (Venakides 1985) where it was rigorously shown for decaying initial data that the Whitham modulation equations describe the limiting solution. The exact solution to the KdV 1-phase Whitham equations for non-breaking initial data was constructed in (Tian 1994) using a generalized hodograph transformation (Tsarev 1985). The case of suitable breaking initial data for the KdV 1-phase Whitham equations was considered in (Grava, Tian 2002) where it was shown that multi-phase behavior is extinguished in finite time.

Various perturbations to and applications of the KdV equation in the small dispersion regime have been considered. The KdV-Burgers equation, which incorporates weak dissipation, was studied in (Gurevich, Pitaevskii 1987). Comparison of a perturbed KdV equation with shallow water undular bore formation on the Australian North West Shelf was performed in (Smyth, Holloway 1988). A variable coefficient KdV equation leading to interacting rarefaction and dispersive shock waves was studied in (El, Grimshaw 2002). Singular bubble pinch-off in Hele-Shaw flow was regularized by considering the KdV equation with small dispersion (Teodorescu, Zabrodin, Wiegmann 2005), (Bettelheim et al. 2005), (Lee, Bettelheim, Wiegmann 2006), (Lee, Teodorescu, Wiegmann 2009). Using a general technique developed for perturbed integrable equations (Kamchatnov 2004), undular bores in a perturbed KdV equation describing the effects of slowly varying topography and bottom friction in shallow water flows were studied in (El, Grimshaw, Kamchatnov 2007). The Whitham modulation equations for the defocusing complex modified KdV equation were shown to be neither strictly hyperbolic nor genuinely nonlinear yielding a complex classification of solutions to the dispersive Riemann problem (Kodama, Pierce, Tian 2008). All possible nonlinear wave interactions resulting from two-step, piecewise constant initial data for the KdV equation in the small dispersion regime were classified in (Ablowitz, Baldwin, Hoefer 2009). Cases include transient, 2-phase DSW interactions and the generation of a DSW or rarefaction wave accompanied by a finite number of isolated solitons.

Solutions to the dispersive Riemann problem can be used to study other hydrodynamic-type problems for the NLS equation. With application to ultrashort pulses in nonlinear optics, ref. (Biondini, Kodama 2006) considered piecewise constant initial data for the NLS equation and, using what is termed here the global regularization method, showed how multi-phase interactions can arise. This method was exploited further in (Hoefer, Ablowitz 2007) to describe the 2-phase interactions of NLS DSWs. Modulated 1-phase interactions of NLS degenerate rarefaction waves were studied in (Hoefer, Engels, Chang 2008) with application to nonlinear quantum interference of matter waves. Two-dimensional DSWs as soliton trains in NLS flow past an obstacle were described in (El, Kamchatnov 2006). The leading and trailing edge behavior associated with optical DSWs propagating in a photorefractive medium, a generalization of the NLS equation, were derived in (El et al. 2007). Dispersive shock waves in a nonlocal NLS type equation were studied both theoretically and experimentally in (Ghofraniha et al. 2007) and (Barsi et al. 2007). Shock waves in discrete nonlinear Schrodinger systems were studied theoretically in (Konotop, Salerno 1997) and observed experimentally in (Jia, Wan, Fleischer 2007). The flow generated by an impenetrable barrier for the NLS equation (the dispersive piston problem) was asymptotically solved in (Hoefer, Ablowitz, Engels 2008). The NLS flow through a penetrable barrier was studied in (El et al. 2009). The Whitham averaging theory was first applied in the context of BECs in (Kamchatnov, Gammal, Kraenkel 2004). A method to generate DSWs in a BEC using Feshbach resonances leading to a time varying nonlinear coefficient in the 2D and 3D NLS equations was proposed in (Perez-Garcia, Konotop, Brazhnyi 2004).

While the dispersive Riemann problem for the defocusing NLS equation (19) leads to the formation of 1-phase DSWs and/or rarefaction waves, the focusing NLS equation $\tag{23} i \varepsilon \psi_t = -\frac{\varepsilon^2}{2} \psi_{xx} - |\psi|^2 \psi ,$

where the sign of the nonlinear term is negative, in the small dispersion limit $$\varepsilon \to 0$$ leads to very different behavior. Modulational instability (Benjamin, Feir 1967) implies that the constant, plane wave background for the dispersive Riemann problem is unstable leading to the rapid growth of any small perturbations. The Whitham modulation equations are of the elliptic type (Forest, Lee 1986) and therefore the initial value problem is ill posed. Numerical results (Miller, Kamvissis 1998), (Ceniceros, Tian 2002), (Cai, McLaughlin, McLaughlin 2002), (Lyng, Miller 2007) for localized initial data reveal a cascade of nonlinear caustics or breaking curves where the solution behavior changes from smooth to oscillatory and then to multi-phase. This problem has been studied rigorously for a specific class of localized initial conditions (Kamvissis, McLaughlin, Miller 2003), (Tovbis, Venakides, Zhou 2004, 2006, 2007) and also for two colliding plane wave (Buckingham, Venakides 2007) initial conditions. Dispersive shock waves in attractive Bose-Einstein condensates (Abdullaev et al. 2005) and nematic liquid crystals (Assanto, Marchant, Smyth 2008) have been modeled by considering eq. (23) and a nonlocal extension of it, respectively.

# References

• F. Kh. Abdullaev, A. Gammal, A. M. Kamchatnov, L. Tomio, Dynamics of bright matter wave solitons in a Bose-Einstein condensate, Int. J. Mod. Phys. B, 19:3415-3473 (2005).
• M. J. Ablowitz, D. E. Baldwin, M. A. Hoefer, Soliton generation and multiple phases in dispersive shock and rarefaction wave interaction, Phys. Rev. E, 80:016603 (2009).
• M. J. Ablowitz, D. J. Benney, The evolution of multi-phase modes for nonlinear dispersive waves, Stud. Appl. Math., 49:225-238 (1970).
• M. J. Ablowitz and H. Segur, Solitons and the inverse scattering transform, SIAM, Philadelphia (1981).
• G. Assanto, T. R. Marchant, and N. F. Smyth, Collisionless shock resolution in nematic liquid crystals, Phys. Rev. A, 78:063808 (2008).
• C. Barsi, W. Wan, C. Sun, and J. W. Fleischer, Dispersive shock waves with nonlocal nonlinearity, Opt. Lett., 32:2930 (2007).
• C. Barsi, W. Wan, J. W. Fleischer, Imaging through nonlinear media using digital holography, Nat. Photonics, 3:211 (2009).
• P. M. Bazin, La propagation des ondes, Mem. Pres. Acad. Sci., Paris 19:495 (1850).
• T. B. Benjamin and J. F. Feir, The disintegration of wave trains on deep water, J. Fluid Mech. 27:417-430 (1967).
• E. Bettelheim, O. Agam, A. Zabrodin, P. Wiegmann, Singular limit of Hele-Shaw flow and dispersive regularization of shock waves, Phys. Rev. Lett., 95:244504 (2005).
• G. Biondini and Y. Kodama, On the Whitham equations for the defocusing nonlinear Schrodinger equation with step initial data, J. Nonlin. Sci., 16:435-481 (2006).
• A. M. Bloch and Y. Kodama, Dispersive reglarization of the Whitham equation for the Toda lattice, SIAM J. Appl. Math., 52:909-928 (1992).
• R. Buckingham and S. Venakides, Long-time asymptotics of the nonlinear Schrodinger equation shock problem, Comm. Pure Appl. Math., 60:1349-1414 (2007).
• D. Cai, D. W. McLaughlin, K. T. R. McLaughlin, The nonlinear Schrodinger equation as both a PDE and a dynamical system in Handbook of dynamical systems, 2:599-675, Amsterdam:North-Holland (2002).
• H. D. Ceniceros and F.-R. Tian, A numerical study of the semi-classical limit of the focusing nonlinear Schrodinger equation, Phys. Lett. A, 306:25-34 (2002).
• J. J. Chang, P. Engels, and M. A. Hoefer, Formation of dispersive shock waves by merging and splitting Bose-Einstein condensates, Phys. Rev. Lett., 101:170404 (2008).
• T. Claeys and T. Grava, Universality of the break-up profile for the KdV equation in the small dispersion limit using the Riemann-Hilbert approach, Comm. Math. Phys., 286:979-1009 (2009).
• T. Claeys and T. Grava, Painlevé II asymptotics near the leading edge of the oscillatory zone for the Korteweg - de Vries equation in the small-dispersion limit, Comm. Pure Appl. Math., early view online (2009).
• C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, Observation of a gradient catastrophe generating solitons, Phys. Rev. Lett., 102:083902 (2009).
• B. Dubrovin, On Hamiltonian perturbations of hyperbolic systems of conservation laws, II: Universality of critical behavior, Comm. Math. Phys., 267:117-139 (2006).
• Z. Dutton, M. Budde, C. Slowe, and L. V. Hau, Observation of quantum shock waves created with ultra-compressed slow light pulses in a Bose-Einstein condensate, Science, 293:663-668 (2001).
• G. A. El, V. V. Geogjaev, A. V. Gurevich, and A. L. Krylov, Decay of an initial discontinuity in the defocusing NLS hydrodynamics, Physica D, 87:186-192 (1995).
• G. A. El and R. H. J. Grimshaw, Generation of undular bores in the shelves of slowly-varying solitary waves, Chaos, 12:1015-1026 (2002).
• G. A. El, Resolution of a shock in hyperbolic systems modified by weak dispersion, Chaos, 15:037103 (2005).
• G. A. El and A. M. Kamchatnov, Spatial dispersive sock waves generated in supersonic flow of Bose-Einstein condensate past slender body, Phys. Lett. A, 350:192-196 (2006).
• G. A. El, A. Gammal, E. G. Khamis, R. A. Kraenkel, and A. M. Kamchatnov, "Theory of optical dispersive shock waves in photorefractive media", Phys. Rev. A, 76:053813 (2007).
• G. A. El, R. H. J. Grimshaw, and A. M. Kamchatnov, Evolution of solitary waves and undular bores in shallow-water flows over a gradual slope with bottom friction, J. Fluid Mech., 585:213:244 (2007).
• H. Flaschka, M. G. Forest, and D. W. McLaughlin, Multiphase averaging and the inverse spectral solution of the Korteweg-de Vries equation, Comm. Pure Appl. Math., 33:739-784 (1980).
• M. G. Forest, J. E. Lee, Geometry and modulation theory for the periodic nonlinear Schrodinger equation in Oscillation theory, computation and methods of compensated compactness, Springer, NY (1986).
• N. Ghofraniha, C. Conti, G. Ruocco, and S. Trillo, Shocks in nonlocal media, Phys. Rev. Lett., 99:043903 (2007).
• T. Grava and F.-R. Tian, The generation, propagation, and extinction of multiphases in the KdV zero-dispersion limit, Comm. Pure Appl. Math., 55:1569-1639 (2002).
• T. Grava and C. Klein, Numerical solution of the small dispersion limit of Korteweg - de Vries and Whitham equations, Comm. Pure Appl. Math., 60:1623-1664 (2007).
• T. Grava and C. Klein, Numerical study of a multiscale expansion of the Korteweg–de Vries equation and Painlevé-II equation, Proc. Roy. Soc. A, 464:733-757 (2008).
• A. V. Gurevich and L. P. Pitaevskii, Nonstationary structure of a collisionless shock wave, Sov. Phys. JETP, 38:291-297 (1974).
• A. V. Gurevich and A. L. Krylov, Dissipationless shock waves in media with positive dispersion, Sov. Phys. JETP, 65:944-953 (1987).
• A. V. Gurevich and L. P. Pitaevskii, Averaged description of waves in the Korteweg-deVries-Burgers equation, Sov. Phys. JETP, 66:490-495 (1987).
• M. A. Hoefer, M. J. Ablowitz, I. Coddington, E. A. Cornell, P. Engels, and V. Schweikhard, Dispersive and classical shock waves in Bose-Einstein condensates and gas dynamics, Phys. Rev. A, 74:023623 (2006).
• M. A. Hoefer and M. J. Ablowitz, Interactions of dispersive shock waves, Physica D, 236:44-64 (2007).
• M. A. Hoefer, M. J. Ablowitz, and P. Engels, Dispersive piston problem, Phys. Rev. Lett., 100:084504 (2008).
• M. A. Hoefer, P. Engels, and J. J. Chang, Matter-wave interference in Bose-Einstein condensates: A dispersive hydrodynamic perspective, Physica D, (2008), doi:10.1016/j.physd.2008.08.021.
• S. Jia, W. Wan, and J. W. Fleischer, Dispersive shock waves in nonlinear arrays, Phys. Rev. Lett., 99:223901 (2007).
• A. M. Kamchatnov, On Whitham theory for perturbed integrable equations, Physica D, 188:247 (2004).
• A. M. Kamchatnov, A. Gammal, and R. A. Kraenkel, Dissipationless shock waves in Bose-Einstein condensates with repulsive interaction between atoms, Phys. Rev. A, 69:063605 (2004).
• S. Kamvissis, K. T. R. McLaughlin, P. D. Miller, Semiclassical soliton ensembles for the focusing nonlinear Schrodinger equation, Ann. Math. Stud., 154, Princeton University Press, Princeton, NJ (2003).
• E. Ya. Khruslov, Splitting of an initial step-like perturbation of the KdV equation, Pis'ma Zh. Eksper. Teoret. Fiz. 21:469-472 (1975); English transl. in JETP Lett. 21:217 (1975).
• E. Ya. Khruslov, Asymptotics of the solution of the Cauchy problem for the Korteweg-de Vries equation with step-like initial data, Mat. Sb., 99:261-281 (1975); English transl. in Math. USSR-Sb. 28:229-248 (1976).
• Y. Kodama, The Whitham equations for optical communications: mathematical theory of NRZ, SIAM J. Appl. Math, 59:2162-2192 (1999).
• Y. Kodama, V. U. Pierce, F.-R. Tian, On the Whitham equations for the defocusing complex modified KdV equation, SIAM J. Math. Anal., 41:26-58 (2008).
• V. V. Konotop and M. Salerno, Dark and bright shock waves on oscillating backgrounds in a discrete nonlinear Schrodinger equation, Phys. Rev. E, 56:3611-3618 (1997).
• P. D. Lax and C. D. Levermore, The small dispersion limit of the Korteweg-deVries equation 1-3, Comm. Pure Appl. Math., 36:253-290,571-593,809-830 (1983).
• S.-Y. Lee, E. Bettelheim, P. Wiegmann, Bubble break-off in Hele-Shaw flows: singularities and integrable structures, Physica D, 219:22 (2006).
• S.-Y. Lee, R. Teodorescu, P. Wiegmann, Shocks and finite-time singularities in Hele-Shaw flow, Physica D, 238:1113-1128 (2009).
• A. M. Leszczyszyn, G. A. El, Y. G. Gladush, and A. M. Kamchatnov, Transcritical flow of a Bose-Einstein condensate through a penetrable barrier, Phys. Rev. A, 79:063608 (2009).
• D. Levermore, The hyperbolic nature of the zero dispersion KdV limit, Commun. Part. Diff. Eq., 13:495-514 (1988).
• G. D. Lyng and P. D. Miller, The N-soliton of the focusing nonlinear Schrodinger equation for N large, Comm. Pure Appl. Math., 60:951-1026 (2007).
• P. D. Miller and S. Kamvissis, On the semiclassical limit of the focusing nonlinear Schrödinger equation, Phys. Lett. A, 247:75-86 (1998).
• M. V. Pavlov, Nonlinear Schrödinger equation and the Bogolyubov-Whitham method of averaging, Teor. Mat. Fiz., 71:351-356 (1987).
• V. M. Perez-Garcia, V. V. Konotop, V. A. Brazhnyi, Feshbach resonance induced shock waves in Bose-Einstein condensates, Phys. Rev. Lett., 92:220403 (2004).
• J. E. Rothenberg and D. Grischkowsky, Observation of the formation of an optical intensity shock and wave breaking in the nonlinear propagation of pulses in optical fibers, Phys. Rev. Lett., 62:531-534 (1989).
• T. P. Simula, P. Engels, I. Coddington, V. Schweikhard, E. A. Cornell, and R. J. Ballagh, Observations on sound propagation in rapidly rotating Bose-Einstein condensates, Phys. Rev. Lett., 94:080404 (2005).
• N. F. Smyth and P. E. Holloway, Hydraulic jump and undular bore formation on a shelf break, J. Phys. Oceanog., 18:947-962 (1988).
• R. J. Taylor, D. R. Baker, and H. Ikezi, Observation of collisionless electrostatic shocks, Phys. Rev. Lett., 24:206-209 (1970).
• R. Teodorescu, A. Zabrodin, P. Wiegmann, Unstable fingering patterns of Hele-Shaw flows as a dispersionless limit of the KdV hierarchy, Phys. Rev. Lett. 95:044502 (2005).
• F.-R. Tian, The initial value problem for the Whitham averaged system, Commun. Math. Phys., 166:79-115 (1994).
• A. Tovbis, S. Venakides, X. Zhou, On semiclassical (zero dispersion limit) solutions of the focusing nonlinear Schrodinger equation, Comm. Pure Appl. Math., 57:877-985 (2004).
• A. Tovbis, S. Venakides, X. Zhou, On the long-time limit of semiclassical (zero dispersion limit) solutions of the focusing nonlinear Schrodinger equation: pure radiation case, Comm. Pure Appl. Math., 59:1379-1432 (2006).
• A. Tovbis, S. Venakides, X. Zhou, Semiclassical focusing nonlinear Schrodinger equation I: inverse scattering map and its evolution for radiative initial data, Int. Math. Res. Notices, 2007:rnm094 (2006).
• S. P. Tsarev, Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type, Sov. Math. Dokl. 31:488-491 (1985).
• S. Venakides, The zero-dispersion limit of the Korteweg-de Vries equation with non-trivial reflection coefficient, Comm. Pure Appl. Math., 38:125-155 (1985).
• W. Wan, S. Jia, and J. W. Fleischer, Dispersive superfluid-like shock waves in nonlinear optics, Nat. Phys., 3:46-51 (2007).
• G. Whitham, Non-linear dispersive waves, Proc. Roy. Soc. Ser. A, 283:238-261 (1965).
• G. Whitham, Linear and nonlinear waves, Wiley, NY (1974).

Internal references

• Tomasz Downarowicz (2007) Entropy. Scholarpedia, 2(11):3901.
• Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
• Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
• David H. Terman and Eugene M. Izhikevich (2008) State space. Scholarpedia, 3(3):1924.