work made clear the reason why Whitham’s equations could be placed in Riemann invariant form; it is a consequence of integrability.)

The task that remained in the use of integrable machinery to study the zero-dispersion limit of the KdV equation was to rigorously analyze the Cauchy problem using the inverse-scattering transform. The first step in this program was taken by Lax and Levermore [@Lax; @1983-1] who considered positive initial data $v_0(x)$ rapidly decaying to zero for large $|x|$. They used WKB methods to argue that the Schrödinger operator with potential $-v_0(x)$ that arises in the scattering theory is approximately reflectionless in the limit $\epsilon\downarrow 0$. On an *ad-hoc* basis they replaced the true scattering data by its WKB analogue, retaining only contributions from a set of $N(\epsilon)\sim
\epsilon^{-1}$ discrete eigenvalues which are approximated by a Bohr-Sommerfeld quantization rule, which amounts to replacing the solution $v_\epsilon(x,t)$ of the Cauchy problem with another solution $\tilde{v}_\epsilon(x,t)$ of having $\epsilon$-dependent initial data close to $v_0$. In this situation, the inverse-scattering procedure reduces to finite-dimensional (of dimension $N(\epsilon)$) linear algebra, and in fact the solution obtained by Cramer’s rule can be reduced to the determinantal formula $$\tilde{v}_\epsilon(x,t)=
2\epsilon^2\frac{\partial^2}{\partial x^2}\log(\tau(x,t)),\quad
\tau(x,t)=\det(\mathbb{I}+{\ensuremath{\mathbf{G}}}(x,t)),$$ where ${\ensuremath{\mathbf{G}}}(x,t)$ is a positive-definite 