different from that of the KdV equation. In particular, the nonlocality in the equation due to the presence of the Hilbert transform is mirrored in a certain nonlocality of the scattering and inverse-scattering problems. In place of the spectral theory of the selfadjoint Schrödinger (Sturm-Liouville) differential operator one has to work with the spectral theory of the nonlocal operator $$\mathcal{L}:=-i\epsilon\frac{\partial}{\partial x} - \mathcal{C}_+
\circ {u_\epsilon}\circ \mathcal{C}_+,\quad
\mathcal{C}_+[f](x):=\lim_{\delta\downarrow 0}\frac{1}{2\pi i}
\int_\mathbb{R}\frac{f(y)}
{y-x-i\delta}\,dy.
\label{eq:Loperator}$$ Here, the operator $\mathcal{C}_+$ is the selfadjoint orthogonal projection from $L^2(\mathbb{R})$ onto the Hardy space of the upper half-plane, the Hilbert space on which $\mathcal{L}$ is selfadjoint, and ${u_\epsilon}$ denotes the operator of multiplication by ${u_\epsilon}(\cdot,t)$.

Certainly a key step forward in the theory of the zero-dispersion limit was taken by Dobrokhotov and Krichever [@Dobrokhotov; @1991] who noted that the second (time evolution) equation in the Lax pair for the BO equation (see below) is simply a time-dependent Schrödinger equation whose potential is a function with an analytic continuation from the real $x$-axis into the upper half-plane and were able to adapt a pre-existing construction of “integrable” potentials for this equation to the appropriate Hardy-space setting. This allowed them to construct, from the Lax pair, a large family 