Long range scattering for nonlinear Schrödinger equations in one space dimension

We consider the scattering problem for the nonlinear Schrödinger equation in 1+1 dimensions:[Figure not available: see fulltext.] where ∂ = ∂/∂x, λ∈R{set minus}{0}, μ∈R, p>3. We show that modified wave operators for (*) exist on a dense set of a neighborhood of zero in the Lebesgue space L²(R) or in the Sobolev space H¹(R)., The modified wave operators are introduced in order to control the long range nonlinearity λ|u|²u.