of inverse scattering, $(\rho ,R)$ are commonly referred to as the (right) scattering data[^3].

In the context of the Cauchy problem for the KdV equation, we have a two parametric family of Marchenko operators $\mathbb{M}(z,t)$, where $(z,t)$ are the (real) variables in . Namely, the measure $\rho _{z,t}$ and the symbol $R_{z,t}$ of $\mathbb{M}(z,t)$ are given by $$\begin{aligned}
d\rho _{z,t}(\lambda )& =\zeta _{z,t}(i\lambda )d\rho (\lambda ), \\
R_{z,t}(\lambda )& =\zeta _{z,t}(\lambda )R(\lambda ), \\
\zeta _{z,t}(\lambda )& :\equiv e^{8i\lambda ^{3}t+2i\lambda z}\end{aligned}$$and $(\rho ,R)$ are the scattering data for the profile $q$. Clearly for $\lambda \in \mathbb{R}$ $$\begin{aligned}
\left\vert \zeta _{z,t}(\lambda )\right\vert & =1, \\
\left\vert \zeta _{z,t}(i\lambda )\right\vert & =e^{8\lambda ^{3}t-2\lambda
z}.\end{aligned}$$

The whole point of the IST is that $(\rho _{0,t},R_{0,t})$ are the scattering data for $H_{u(z,t)}$ where $u(z,t)$ solves . The actual mechanism to recover $u(z,t)$ amounts to solving the Marchenko integral equation[^4] or equivalently through the Riemann-Hilbert problem. For our purposes the explicit formula is convenient. Note that is nothing but Cramer’s rule for linear integral equations. For $R\equiv 0$ (reflectionless initial profile), assuming that $q$ is short-range, $\mathbb{M}(z,t)$ turns into a finite rank operator of a very explicit structure. The formula in this particular case was 