21st Century COE Program : Mathematics of Nonlinear Structure via Singularity

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.

21st Century COE Program : Mathematics of Nonlinear Structure via Singularity