Well-posedness for the nonlocal nonlinear Schrödinger equation

We establish local well-posedness for small   initial data in the usual Sobolev spaces Hs(R)Hs(R), s⩾1s⩾1, and global well-posedness in H1(R)H1(R), for the Cauchy problem associated to the nonlocal nonlinear Schrödinger equation∂tu=−iα∂x2u+βu∂x(|u|2)−iβuTh∂x(|u|2)+iγ|u|2u, where u=u(x,t)u=u(x,t), x,t∈Rx,t∈R, ThTh is a singular integral operator, α>0α>0, β⩾0β⩾0 and γ⩾0γ⩾0 are real constants. Our method of proof is based on the smoothing effects produced by the linear Schrödinger equation.