  We consider the question of global existence of small, smooth, and localized
solutions of a certain fractional semilinear cubic NLS in one dimension,
$$i\partial_t u - \Lambda u = c_0{|u|}^2 u + c_1 u^3 + c_2 u \bar{u}^2 + c_3
\bar{u}^3, \qquad \Lambda = \Lambda(\partial_x) = {|\partial_x|}^(1/2)$$, 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.
