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

TL;DR

This paper investigates the well-posedness of a generalized derivative nonlinear Schrödinger equation, establishing local solutions in Sobolev spaces and discussing conditions for global existence.

Contribution

It proves local well-posedness in H^1 and H^2, and constructs solutions without compactness, advancing understanding of this nonlinear PDE.

Findings

Local well-posedness in H^1 and H^2

Solutions constructed as limits of approximate solutions

Discussion on global existence in H^1

Abstract

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 $H^1$.