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Existence of excited states for a nonlinear Dirac field

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Abstract

We prove the existence of infinitely many stationary states for the following nonlinear Dirac equation

$$i\gamma ^\mu \partial _\mu \psi - m\psi + (\bar \psi \psi )\psi = 0.$$

Seeking for eigenfunctions splitted in spherical coordinates leads us to analyze a nonautonomous dynamical system inR 2. The number of eigenfunctions is given by the number of intersections of the stable manifold of the origin with the curve of admissible datum. This proves the existence of infinitely many stationary states, ordered by the number of nodes of each component.

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Authors and Affiliations

  1. CSP, Université Paris-Nord, Avenue J. B. Clément, F-93430, Villetanneuse, France

    Mikhael Balabane

  2. CMA, Ecole Normale Supérieure, 45, rue d'Ulm, F-75230, Paris Cedex 05, France

    Mikhael Balabane & Frank Merle

  3. Analyse Numérique, Université Pierre et Marie Curie, 4, place Jussieu, F-75252, Paris Cedex 05, France

    Thierry Cazenave

  4. Département de Mathématiques, Université Paris-Sud, F-91405, Orsay Cedex, France

    Adrien Douady

  5. Ecole Normale Supérieure, 45, rue d'Ulm, F-75230, Paris Cedex 05, France

    Adrien Douady

Authors
  1. Mikhael Balabane
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  2. Thierry Cazenave
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  3. Adrien Douady
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  4. Frank Merle
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Communicated by B. Simon

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Balabane, M., Cazenave, T., Douady, A. et al. Existence of excited states for a nonlinear Dirac field. Commun.Math. Phys. 119, 153–176 (1988). https://doi.org/10.1007/BF01218265

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  • DOI: https://doi.org/10.1007/BF01218265

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