## Jean Ginibre : Nonlinear Dispersive Equations, Invited Lecture

### Long range scattering for some Schrödinger related nonlinear systems

### Abstract

We reconsider the theory of scattering and more precisely the existence of
modified wave operators for the Wave-Schrödinger system (WS)_{3} and for the
Maxwell-Schrödinger system (MS)_{3} in space dimension 3, where both systems
belong to the limiting long range case. Improving and simplifying an available
direct method (which requires smallness of the Schrödinger data), we prove
the existence of modified wave operators without any size restriction on the
Wave or Maxwell field, thereby improving previous results. The method exploits
systematically the Strichartz inequalities for the Schrödinger equation and
for the Wave equation. It applies also to the Klein-Gordon-Schrödinger system
(KGS)_{2} in space dimension 2 and to the Zakharov system (Z)_{3} in space
dimension 3 (the latter is short range). In the last two cases, the Strichartz
inequalities are needed only for the Schrödinger equation.