Local well-posedness for quadratic Schr\"odinger equations in $\mathbf{R^{1+1}}$: a normal form approach

For the Schr\"odinger equation $u_t+i u_{xx}=\nab^\be[u^2]$, $\be\in (0,1/2)$, we establish local well-posedness in $H^{\be-1+}$ (note that if $\be=0$, this matches, up to an endpoint, the sharp result of Bejenaru-Tao, \cite{BT}). Our approach differs significantly from the previous ones in that we use normal form transformation to analyze the worst interacting terms in the nonlinearity and then show that the remaining terms are (much) smoother. In particular, this allows us to conclude that $u-e^{-i t \p_x^2} u(0)\in H^{-\f{1}{2}}(\rone)$, even though $u(0)\in H^{\be-1+}$. In addition and as a byproduct of our normal form analysis, we obtain a Lipschitz continuity property in $H^{-\f{1}{2}}$ of the solution operator (which originally acts on $H^{\be-1+}$), which is new even in the case $\be=0$. As an easy corollary, we obtain local well-posedness results for $u_t+ i u_{xx} = [\nab^{\beta} u]^2$. Finally, we sketch an approach to obtain similar statements for the equations $u_t+i u_{xx}=\nab^\be[u\bar{u}]$ and $u_t+i u_{xx}=\nab^\be[\bar{u}^2]$.