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Wave operators and analytic solutions for systems of non-linear Klein-Gordon equations and of non-linear Schrödinger equations

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Abstract

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 non-linear Klein-Gordon equations all masses are supposed to be different from zero.

We prove, for such systems, that the wave operator (fromt=∞ tot=0) exists on a domain of small entire test functions of exponential type and that the analytic Cauchy problem, in ℝ+×ℝ3, has a unique solution for each initial condition (att=0) being in the image of the wave operator. The decay properties of such solutions are discussed in detail.

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Authors and Affiliations

  1. Department of Mathematical Physics, University of Dijon, BP 138, F-21004, Dijon Cedex, France

    Jacques C. H. Simon & Erik Taflin

Authors
  1. Jacques C. H. Simon
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  2. Erik Taflin
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Additional information

Communicated by H. Araki

Partially supported by the Swiss National Science Foundation

On leave from Institut de Physique Théorique, 32 Bd d'Ivoy, CH-1211 Geneve 4 Switzerland.

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Simon, J.C.H., Taflin, E. Wave operators and analytic solutions for systems of non-linear Klein-Gordon equations and of non-linear Schrödinger equations. Commun.Math. Phys. 99, 541–562 (1985). https://doi.org/10.1007/BF01215909

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  • DOI: https://doi.org/10.1007/BF01215909

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