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 $\text{deg }P\ge 3$, 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 $\text{deg }P\ge 5$, the solution can be extended globally. By restricting to equations of the form $i{\mathrm{\partial }}_{t}u+\mathrm{\Delta }u={\mathrm{\partial }}_{x}P(u,\bar{u})$ with $$\text{deg }$ P\geq5$, we were able to obtain the global well-posedness in the critical Sobolev space.