Nonlinear scattering theory for a class of wave equations in Hs

For the semi-linear (higher order) wave equation and the nonlinear (higher order) Schrödinger equation, we show that the scattering operators map a band in Hs into Hs if the nonlinearities have (sub-)critical powers in Hs . The smoothness of the scattering operators and the uniform boundedness of strong solutions for the defocusing NLS equation are also shown provided that the nonlinearities have subcritical growth in H1. Moreover, the spatial decaying behavior of solutions in energy space for the defocusing NLS equation are obtained.  2004 Elsevier Inc. All rights reserved.