A linear Schrödinger approximation for the KdV equation via IST beyond the natural NLS time scale

We are interested in improving validity results for the Nonlinear Schrödinger (NLS) approximation beyond the natural time scale for completely integrable systems. As a first step, we consider this approximation for the Korteweg-de Vries (KdV) equation with initial conditions for which the scattering data contains no eigenvalues. By performing a linear Schrödinger approximation for the scattering data the error made by this approximation has only to be estimated for a purely linear problem which gives estimates beyond the natural NLS time scale. The inverse scattering transform allows us to transfer these estimates to the original variables.