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

Ioan Bejenaru, Terence Tao

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…