Global well-posedness for the higher order non-linear Schrödinger equation on modulations spaces

Abstract

We consider the initial value problem (IVP) associated to a higher order nonlinear Schrödinger (h-NLS) equation ${\partial }_{t}u+ia{\partial }_x^{2}u+b{\partial }_x^{3}u+i{c}_1|u{|}^{2}u+c_2|u{|}^2{\partial }_{x}u=0,x,t\in \mathbb{R},$ for given data in the modulation space $M_s^{2,p}(\mathbb{R})$. Using ideias of Killip, Visan, Zhang, Oh, Wang, we prove that the IVP associated to the h-NLS equation is globally well-posed in the modulation spaces $M^{s,p}$ for $s\ge \frac{1}{4}$ and $p\ge 2$.