Sharp global well-posedness for the cubic nonlinear Schr\"odinger equation with third order dispersion

We consider the initial value problem (IVP) associated to the cubic nonlinear Schr\"odinger equation with third-order dispersion \begin{equation*} \partial_{t}u+i\alpha \partial^{2}_{x}u- \partial^{3}_{x}u+i\beta|u|^{2}u = 0, \quad x,t \in \mathbb{R}, \end{equation*} for given data in the Sobolev space $H^s(\mathbb{R})$. This IVP is known to be locally well-posed for given data with Sobolev regularity $s>-\frac14$ and globally well-posed for $s\geq 0$ [3]. For given data in $H^s(\mathbb{R})$, $0>s> -\frac14$ no global well-posedness result is known. In this work, we derive an almost conserved quantity for such data and obtain a sharp global well-posedness result. Our result answers the question left open in [3].