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

We consider the initial value problem (IVP) associated to a higher order nonlinear Schrödinger (h-NLS) equation ${\mathrm{\partial }}_{t}u+ia{\mathrm{\partial }}_x^{2}u+b{\mathrm{\partial }}_x^{3}u+i{c}_1\mathrm{\mid }u{\mathrm{\mid }}^{2}u+c_2\mathrm{\mid }u{\mathrm{\mid }}^2{\mathrm{\partial }}_{x}u=0,x,t\in \mathbb{R},$ 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 $p\ge 2$.