real symmetric matrix of dimension $N(\epsilon)\times N(\epsilon)$. Lax and Levermore then established the existence of the limit of $2\epsilon^2\log(\tau(x,t))$ as $\epsilon\downarrow 0$, uniformly on compact subsets of the $(x,t)$-plane. This yields weak convergence of $\tilde{v}_\epsilon(x,t)$ by differentiation of the limit function with respect to $x$. The Lax-Levermore method is to expand the determinant $\tau(x,t)$ in principal minors indexed by subsets of the set of eigenvalues; noting that each term is positive they showed that the sum of terms is asymptotically dominated by its largest term, and they further approximated this discrete optimization problem with an $\epsilon$-independent (limiting) convex variational problem, explicitly parametrized by $x$ and $t$, for measures. The weak zero-dispersion limit of the Cauchy problem for the KdV equation is therefore encoded implicitly in the solution of this variational problem. The Lax-Levermore method reproduces the specified initial data $v_0(x)$ at $t=0$ as $\epsilon\downarrow 0$, which establishes validity, in a certain sense, of the WKB-based spectral approximation procedure in the first step.

Later, Venakides [@Venakides90] was able to extend the method of Lax and Levermore to higher order, capturing the form of the oscillations that are averaged out in the weak limit. This work at last made clear that the 