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∂tu−Λu=c0|u|2u+c1u3+c2uu¯2+c3u¯3,Λ=Λ(∂x)=|∂x|12, where c0∈R and c1,c2,c3∈C. This model is motivated by the two-dimensional water wave equation, which has 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.