The inverse scattering transform for the KdV equation with step-like singular Miura initial profiles

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 $t>0$.