Norm inflation in negative order Sobolev spaces for KdV and KP

We prove norm inflation phenomena for KdV and KP equations in negative order Sobolev spaces, in the periodic case, as well as on the whole space, on an arbitrarily large scale of negative order Sobolev spaces as target spaces. The proof relies on WKB analysis for a semiclassical version of the equation, in a weakly nonlinear r{\'e}gime, and the creation of the zero Fourier mode by resonant interaction. Unlike in previous similar results, this average mode has a smaller order of magnitude than the initial data, which requires a more detailed WKB analysis.