Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation

We establish that the quadratic non-linear Schrödinger equation iut+uxx=u2,iut+uxx=u2,where u:R×R→Cu:R×R→C, is locally well-posed in Hs(R)Hs(R) when s⩾-1s⩾-1 and ill-posed when s<-1s<-1. Previous work in [C. Kenig, G. Ponce, L. Vega, Quadratic forms for the 1-D semilinear Schrödinger equation, Trans. Amer. Math. Soc. 346 (1996) 3323–3353] had established local well-posedness for s>-34. The local well-posedness is achieved by an iteration using a modification of the standard Xs,bXs,b spaces. The ill-posedness uses an abstract and general argument relying on the high-to-low frequency cascade present in the non-linearity, and a computation of the first non-linear iterate.