initial data $u_0(x)$ in the limit $\epsilon\downarrow 0$ as the *zero-dispersion limit*.

A related problem and its history
---------------------------------

A more famous nonlinear dispersive wave equation is the Korteweg-de Vries (KdV) equation $$\frac{\partial v_\epsilon}{\partial t} + 2v_\epsilon
\frac{\partial v_\epsilon}{\partial x} + 
\frac{1}{3}\epsilon^2 \frac{\partial^3v_\epsilon}{\partial x^3}=0,
\quad x\in\mathbb{R},\quad
t>0,
\label{eq:KdV}$$ a model for long surface waves on shallow water among a wide variety of other physical phenomena. When $\epsilon>0$ is small, this equation displays qualitatively similar behavior to that just illustrated for the BO equation: the dispersive term arrests the shock in the $\epsilon=0$ equation with the formation of a train of waves of amplitude approximately independent of $\epsilon$ and wavelength proportional to $\epsilon$.

The modeling of the zero-dispersion limit for the KdV equation has a long history going back to the work of Whitham [@Whitham65] who used the method of averaging to propose a nonlinear hyperbolic system of three partial differential equations to describe the modulational variables (*e.g.* slowly-varying amplitude, mean, and wavelength of the wavetrain). Whitham noted that the system of modulation equations he obtained had the nongeneric property that by choice of special dependent variables $v^1$, $v^2$, and $v^3$, it could be written in so-called Riemann invariant form, in 