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.
Categories: math.AP