We develop the inverse scattering transform for the KdV equation with real singular initial data $q(x)$ of the form $q(x) = r(x) + r(x)^2$, where $rin L^2_{textrm{loc}}$ and $r=0$ on $mathbb R_+$. As a consequence we show that the solution $q(x,t)$ is a meromorphic function with no real poles for any $t>0$.
Soliton theory and the theory of Hankel (and Toeplitz) operators have stayed essentially hermetic to each other. This paper is concerned with linking together these two very active and extremely large theories. On the prototypical example of the Cauchy problem for the Korteweg-de Vries (KdV) equation we demonstrate the power of the language of Hankel operators in which symbols are conveniently represented in terms of the scattering data for the Schrodinger operator associated with the initial data for the KdV equation. This approach yields short-cuts to already known results as well as to a variety of new ones (e.g. wellposedness beyond standard assumptions on the initial data) which are achieved by employing some subtle results for Hankel operators.
We show that the KdV flow evolves any real singular initial profile q of the form q=r+r^2, where rinL_{loc}^2, r|_{R_+}=0 into a meromorphic function with no real poles.
