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 $q(x)=r^{\mathrm{\prime }}(x)+r(x{)}^2$, where $r\in L^2_{$\text{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 $t>0$.