Long range scattering for some Schr"odinger related nonlinear systems
Résumé
We review some recent results on the theory of scattering and more precisely on the local Cauchy problem at infinity in time for some long range nonlinear systems including some form of the Schr"odinger equation. We consider in particular the Wave-Schr"odinger system in space dimension 3, the Maxwell-Schr"odinger system in space dimension 3, the Klein-Gordon-Schr"odinger system in space dimension 2 and the Zakharov system in space dimensions 2 and 3. By the use of a direct method which is intrinsically restricted to the case of small Schr"odinger data and to the borderline long range case, one can prove the existence of solutions defined for large times and with prescribed asymptotic behaviour in time, without any size restriction on the Wave, Maxwell or Klein-Gordon data. Furthermore one obtains convergence rates of the solutions to their asymptotic forms as negative powers of t in suitable norms.