Modified scattering operator for the derivative nonlinear Schrodinger equation

We consider the derivative nonlinear Schrodinger 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,?+? to the neighborhood of the origin in H1,?, where ? > 1 2 and ? > 0 is small. The weighted Sobolev space is defined by Hm,s = ? L2; (1+x2) s 2 (1-?2x ) m2 ? L2 ? ? .