Well-posedness for the nonlocal nonlinear Schrödinger equation

We establish local well-posedness for small initial data in the usual Sobolev spaces Hs (R), s 1, and global well-posedness in H1(R), for the Cauchy problem associated to the nonlocal nonlinear Schrödinger equation ∂tu=−iα∂2 xu+βu∂x |u|2 −iβuTh∂x |u|2 +iγ |u|2u, where u = u(x, t), x, t ∈ R, Th 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. © 2006 Elsevier Inc. All rights reserved.