Nonlinear fractional Schrodinger equations in one dimension

Abstract

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 derivative(t)u - Lambda u = c(0)vertical bar u vertical bar(2)u + c(1)u(3) + c(2)u (u) over bar (2) + c(3)(u) over bar (3), Lambda = Lambda(partial derivative(x)) = vertical bar partial derivative(x)vertical bar 1/2, where c(0) is an element of R and c(1), c(2), c(3) is an element of 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. (C) 2013 Elsevier Inc. All rights reserved.