It should be pointed out that the weak limit formula is much more explicit than the corresponding formula found by Lax and Levermore [@Lax; @1983-1] for the weak zero-dispersion limit of the Cauchy problem for the KdV equation. Indeed, the latter requires the solution, for each $x$ and $t$, of a constrained functional variational problem, which can be solved in closed form only for the simplest initial data. As we will now see, there are several classical wave propagation problems whose asymptotic behavior can be reduced to the multivalued solution of Burgers’ equation; however, even these simple problems involve more complicated schemes for combining the solution branches than that exhibited in the simple formula .

After we introduce the necessary framework for our study (the inverse-scattering transform for the BO equation) in §\[sec:background\], we will analyze the direct scattering map in the zero-dispersion limit in §\[sec:direct\]. Then we will prove Theorem \[MainTheorem\] in §\[sec:inverse\] by carrying out a detailed analysis of the inverse scattering map applied to the asymptotic formulae for scattering data obtained in §\[sec:direct\]. In §\[sec:corollary\] we prove Corollary \[corr:strong\], and then in §\[sec:numerics\] we will illustrate our results with numerical calculations and address the relation between ${\tilde{u}_\epsilon}(x,t)$ 