Well-posedness for a generalized derivative nonlinear Schr\"odinger equation

We study the Cauchy problem for a generalized derivative nonlinear Schr"odinger equation with the Dirichlet boundary condition. We establish the local well-posedness results in the Sobolev spaces $H^1$ and $H^2$. Solutions are constructed as a limit of approximate solutions by a method independent of a compactness argument. We also discuss the global existence of solutions in the energy space $H^1$.