Well-posedness for the nonlocal nonlinear Schrödinger equation

Abstract

We establish local well-posedness for small initial data in the usual Sobolev spaces H s (R), s 1, and global well-posedness in H 1 (R), for the Cauchy problem associated to the nonlocal nonlinear Schrödinger equationwhere u = u(x, t), x, t ∈ R, T h is a singular integral operator, α > 0, β 0 and γ 0 are real constants. Our method of proof is based on the smoothing effects produced by the linear Schrödinger equation.