Sharp Global Well-Posedness for the Cubic Nonlinear Schrödinger Equation with Third Order Dispersion

Abstract

We consider the initial value problem (IVP) associated to the cubic nonlinear Schrödinger equation with third-order dispersion

$$\begin{aligned} \partial _{t}u+i\alpha \partial ^{2}_{x}u- \partial ^{3}_{x}u+i\beta |u|^{2}u = 0, \quad x,t \in \mathbb R, \end{aligned}$$

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 $s>-\frac{1}{4}$ and globally well-posed for $s\ge 0$ (Carvajal in Electron J Differ Equ 2004:1–10, 2004). For given data in $H^s(\mathbb R)$, $0>s> -\frac{1}{4}$ 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 (Carvajal in Electron J Differ Equ 2004:1–10, 2004).