1 research outputs found
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,
, where
and . 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