On the inverse scattering transform for the KdV equation with summable initial data

Abstract

We consider the Cauchy problem for the Korteweg--de Vries equation with real initial data $q$ that is both $L^1$ and $L^2$ summable and supported on (0,\infty). Using the left reflection coefficient and Hankel operators on the Hardy space $H^2$, we derive a trace-type representation for the corresponding solution. The proof is based on approximation by compactly supported potentials, uniform convergence of the associated reflection coefficients away from the origin, and continuity properties of the resulting Hankel operators. This yields a rigorous inverse scattering construction for a class of summable half-line supported initial data beyond the standard short-range setting.