Modified scattering operator for the derivative nonlinear schrödinger equation

We consider the derivative nonlinear Schrödinger equation i∂tu+ 1 2 ∂2x u = i∂x(|u|2u), t R, x R. Our purpose is to prove that the modified scattering operator is defined as a map from the neighborhood of the origin in H1,α+γ to the neighborhood of the origin in H1,α, where α > 1 2 and γ > 0 is small. The weighted Sobolev space is defined by Hm,s = {φ L2; (1+x2) s 2 (1-∂2x ) m2 φ L2 ≤ ∞}.