View source for Dispersive shock waves

===General Initial Value Problem===
The dispersive Riemann problem considered in Eq. \eqref{E1} 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 problems|initial value problem]]
:<math> u_t + \left( \frac{1}{2} u^2 + \varepsilon^2 u_{xx} \right)_x = 0,</math>
:<math>u(x,0) = u_0(x), \quad \varepsilon^2 \ll 1, </math>
where <math>u_0(x)</math> 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 <math>(u_0'(x) < 0)</math> anywhere, then there is a critical point <math>(x,t) = (x_b,t_b)\ ,</math> the breaking point, where the solution will develop a region of rapid oscillations with wavelength of order <math>\mathcal{O}(\varepsilon)\ .</math> Beyond the breaking point, the oscillatory region was rigorously shown to be described by the Whitham modulation equations in the limit <math>\varepsilon \to 0</math> ([[#lax|Lax, Levermore 1983]]), ([[#venakides85|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 ([[#grava07|Grava, Klein 2007]]) where it was shown that different portions converge to the Whitham modulation solution at different asymptotic rates in <math>\varepsilon\ .</math>
A rigorous asymptotic study in ([[#claeys2009a|Claeys, Grava 2009]]) for <math>(x,t)</math> close to <math>(x_b,t_b)</math> demonstrated that the solution is described by the fourth order ODE
:<math>\label{E4.4}
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 ],
</math>

where <math>X, T, U</math> are rescalings of <math>x, t, u\ .</math>
Further work in ([[#grava2008|Grava, Klein 2008]]) and ([[#claey2009b|Claeys, Grava 2009]]) showed that the leading, soliton edge is asymptotically described by a certain solution to
:<math>
q''(s) = s q + 2 q^3(s) .
</math>

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 ([[#el05|El 2005]]).
However, leading order Whitham theory does not determine the slow variation of the phase <math>\theta_0(x,t)</math> in Eq. \eqref{E4.3} and a generalization to multiple rapidly oscillating phases is required if the Whitham equations themselves develop a gradient catastrophe ([[#flaschka|Flaschka, Forest, McLaughlin 1980]]), ([[#grava|Grava, Tian 2002]]).