Gurevich and Pitaevskii [@Gurevich; @1974] to describe the evolution of a dispersive shock was independently described by Matsuno [@Matsuno; @1998-1; @Matsuno; @1998-2] and by Jorge, Minzoni, and Smyth [@Jorge; @1999]. This matching procedure provides a reasonable approach to the Cauchy problem for the Benjamin-Ono equation with fixed initial data ${u_\epsilon}(x,0)=u_0(x)$ when $\epsilon\ll 1$, but it is based on formal asymptotics. In [@Matsuno; @1998-1], Matsuno writes:

> From a rigorously mathematical point of view, however, the various results presented in this paper should be justified on the basis of an exact method of solution such as \[the inverse-scattering transform\], or an analog of the Lax-Levermore theory for the KdV equation.

It is our intention in this paper to provide exactly such a justification, by developing a new method that does for the BO Cauchy problem exactly what the Lax-Levermore method does for the KdV Cauchy problem.

The main result of our analysis is remarkably easy to state, but first we need to recall some basic facts concerning the equation obtained from simply by setting $\epsilon=0$. Recall that while for general sufficiently smooth initial data $u^\mathrm{B}(x,0)=u_0(x)$ the inviscid Burgers equation $$\frac{\partial u^\mathrm{B}}{\partial t}+2u^\mathrm{B}
\frac{\partial u^\mathrm{B}}{\partial x}=0
\label{eq:inviscidBurgers}$$ does not have a global 