We consider an initial value problem for the Korteweg-de Vries
(KdV) equation. The initial datum belongs to a class of (smooth) functions,
which grow as $\sqrt[3]{-6x}$ as $x\to\pm\infty$. More precisely, the
initial datum is a compactly supported perturbation of the exact solution
$U(x,t)$ of the KdV equation, which simultaneously solves a Painleve-type
equation, namely, the second member of the first Painleve hierarchy. We
develop forward and inverse scattering problem for solving such an initial
value problem. In order to achieve this, we show that the underlying Lax
operator has spectrum, which is a one-folded real line. Furthermore, we
define associated spectral functions $a(\lambda), b(\lambda),$ in terms of
which we formulate an appropriate time-dependent Riemann-Hilbert problem.
The solution of this Riemann-Hilbert problem then generates the solution to
our initial value problem.  This is a joint work with B.Dubrovin, and the
talk is based on arXiv:1901.07470."