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, where u : R × R → C, is locally well-posed in Hs (R) when s − 1 and ill-posed when s <−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 >− 3 4 . The local well-posedness is achieved by an iteration using a modification of the standard Xs,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. © 2005 Elsevier Inc. All rights reserved.