Wave operators and analytic solutions for systems of nonlinear Klein- Gordon equations and of nonlinear Schrödinger equations (Q1820415)

We consider, in a \(1+3\) space time, arbitrary (finite) systems of nonlinear Klein-Gordon equations (respectively Schrödinger equations) with an arbitrary local and analytic non-linearity in the unknown and its first and second order space-time (respectively first order space) derivatives, having no constant or linear terms. No restriction is given on the frequency sign of the initial data. In the case of nonlinear Klein-Gordon equations all masses are supposed to be different from zero. We prove, for such systems, that the wave operator (from \(t=\infty\) to \(t=0)\) exists on a domain of small entire functions of exponential type and that the analytic Cauchy problem, in \({\mathbb{R}}^+\times {\mathbb{R}}^ 3\), has a unique solution for each initial condition (at \(t=0)\) being in the image of the wave operator. The decay properties of such solutions are discussed in detail.