The main tool of soliton theory (aka completely integrable systems) is the inverse scattering transform (IST) which relies on solving the Faddeev-Marchenko integral equation. The latter amounts to inverting the I+Hankel operator which historically was done by classical techniques of integral operators and the theory of Hankel operations was not used. In the recent decade however the interest in the soliton community has started shifting from classical initial conditions of integrable PDEs to more general ones (aka none classical initial data) for which the classical IST no longer works. In this talk, on the prototypical example of the Cauchy problem for the Korteweg-de Vries (KdV) equation, we show how the classical IST can be extended to serve a broad range of physically interesting initial data. Our approach is essentially based on the theory of Hankel operators.