Small data well-posedness for derivative nonlinear Schrödinger equations

We study the local and global solutions of the generalized derivative nonlinear Schrödinger equation i∂tu+Δu=P(u,u‾,∂xu,∂xu‾), where each monomial in P is of degree 3 or higher, in low-regularity Sobolev spaces without using a gauge transformation. Instead, we use a solution decomposition technique introduced in [4] during the perturbative argument to deal with the loss on derivative in nonlinearity. It turns out that when each term in P contains only one derivative, the equation is locally well-posed in H12, otherwise we have a local well-posedness in H32. If each monomial in P is of degree 5 or higher, the solution can be extended globally. By restricting to equations to the form i∂tu+Δu=∂xP(u,u‾) with the quintic nonlinearity, we were able to obtain the global well-posedness in the critical Sobolev space.