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

Abstract

We establish that the quadratic non-linear Schrödinger equation $$i{u}_t+u_{xx}=u^2$$ where $u: \R \times \R \to \C$, is locally well-posed in $H^s(\mathbb{R})$ when $s\ge -1$ and ill-posed when $s<-1$. Previous work of Kenig, Ponce and Vega had established local well-posedness for $s>-3/4$. The local well-posedness is achieved by an iteration using a modification of the standard $X^{s,b}$ 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.

28 pages, no figures, to appear, J.Func. Anal. Some minor gaps filled in