Global well-posedness for the higher order non-linear Schrödinger equation in modulation spaces

Abstract

We consider the initial value problem (IVP) associated with a higher order non-linear Schrödinger (h-NLS) equation${\partial }_{t}u+ia{\partial }_x^{2}u+b{\partial }_x^{3}u=2ia|u{|}^{2}u+6b|u{|}^2{\partial }_{x}u,x,t\in R,$ with given data in the modulation space $M_s^{2,p}\left(R\right)$. Using ideas from Killip, Visan, Zhang, Oh and Wang, we prove that the IVP associated with the h-NLS equation is globally well-posed in the modulation spaces $M_s^{2,p}\left(R\right)$ for $s\ge \frac{1}{4}$ and $p\ge 2$.