Nonlinear fractional Schr\"odinger equations in one dimension

We consider the question of global existence of small, smooth, and localized solutions of a certain fractional semilinear cubic NLS in one dimension, $$i{\mathrm{\partial }}_{t}u-\mathrm{\Lambda }u=c_0{\mathrm{\mid }u\mathrm{\mid }}^{2}u+c_1{u}^3+c_{2}u{\bar{u}}^2+c_3{\bar{u}}^3,\mathrm{\Lambda }=\mathrm{\Lambda }({\mathrm{\partial }}_x)={\mathrm{\mid }{\mathrm{\partial }}_x\mathrm{\mid }}^{(}1\mathrm{/}2)\text{,}$$ where $c_0\in \mathbb{R}$ and $c_1,c_2,c_3\in \mathbb{C}$. This model is motivated by the two-dimensional water waves equations, which have a somewhat similar structure in the Eulerian formulation, in the case of irrotational flows. We show that one cannot expect linear scattering, even in this simplified model. More precisely, we identify a suitable nonlinear logarithmic correction, and prove global existence and modified scattering of solutions.