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Wave operators and analytic solutions for systems of nonlinear Klein- Gordon equations and of nonlinear Schrödinger equations. (English) Zbl 0615.47034

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.

MSC:

47F05 General theory of partial differential operators
35J10 Schrödinger operator, Schrödinger equation
47J25 Iterative procedures involving nonlinear operators
81Q99 General mathematical topics and methods in quantum theory
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