Nonlinear fractional Schrödinger 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, $i\partial_t u - Λu = c_0{|u|}^2 u + c_1 u^3 + c_2 u \bar{u}^2 + c_3 \bar{u}^3, \qquad Λ= Λ(\partial_x) = {|\partial_x|}^(1/2)$, 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.

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