MODIFIED SCATTERING OPERATOR FOR THE DERIVATIVE NONLINEAR SCHRODINGER EQUATION

We consider the derivative nonlinear Schrodinger equation i partial derivative(t)u + 1/2 partial derivative(2)(x)u = i partial derivative(x)(vertical bar u vertical bar(2)u), t is an element of R, x is an element of R. Our purpose is to prove that the modified scattering operator is defined as a map from the neighborhood of the origin in H-1,H-alpha+gamma to the neighborhood of the origin in H-1,H-alpha, where alpha > 1/2 and gamma > 0 is small. The weighted Sobolev space is defined by H-m,H-s = {phi is an element of L-2; parallel to(1 + x(2))(s/2) (1 - partial derivative(2)(x))(m/2) phi parallel to(L2) < infinity}.