Global well-posedness and scattering for the derivative nonlinear Schrödinger equation with small rough data

Abstract

We study the Cauchy problem for the generalized elliptical and non-elliptical derivative nonlinear Schrödinger equations (DNLS) and get the global well posedness of solutions with small data in modulation spaces $M_{2,1}^s({\mathbb{R}}^n)$. Noticing that $B_{2,1}^{s+n/2}\subset M_{2,1}^s\subset B_{2,1}^s$ are optimal inclusions, we have shown the global well posedness of DNLS with a class of rough data. As a by-product, the existence of the scattering operators in modulation spaces with small data is also obtained.