Small data well-posedness for derivative nonlinear Schr\"odinger equations

We study the generalized derivative nonlinear Schr\"odinger equation $i\partial_t u+\Delta u = P(u,\overline{u},\partial_x u,\partial_x \overline{u})$, where $P$ is a polynomial, in Sobolev spaces. It turns out that when $\text{deg } P\geq 3$, the equation is locally well-posed in $H^{\frac{1}{2}}$ when each term in $P$ contains only one derivative, otherwise we have a local well-posedness in $H^{\frac{3}{2}}$. If $\text{deg } P \geq 5$, the solution can be extended globally. By restricting to equations of the form $i\partial_t u+\Delta u = \partial_x P(u,\overline{u})$ with $\text{deg } P\geq5$, we were able to obtain the global well-posedness in the critical Sobolev space.