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Theses

Existence, stabilité et instabilité d'ondes stationnaires pour quelques équations de Klein-Gordon et Schrödinger non linéaires

Abstract : This thesis is devoted to the study of standing waves for nonlinear dispersive equations, in particular the Schrödinger equation but also the Klein-Gordon equation. The works are organized around two main issues : existence and orbital stability of standing waves.

The existence is essentially studied by the way of variational methods. We exhibit various variational characterizations of standing waves, for example as critical points of some functional at the mountain pass level or at the least energy level, or as minimizers of a functional under various constraints.

Depending on the strength of the nonlinearity and on the space dependency, we prove that stability or instability holds for the standing waves. When instability holds, we show that, in some situations, instability occurs by blow up, whereas in other cases the solutions are globally well-posed. In addition to the variational characterization of waves, the study of stability leads us to derive spectral informations. In the first part of this thesis, we show a nondegenerescence result for the linearized operator associated with a limit problem. In the second part, we localize the second eigenvalue of the linearized by the mean of a combinaison of perturbation and continuation arguments.
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https://tel.archives-ouvertes.fr/tel-00239293
Contributor : Stefan Le Coz <>
Submitted on : Tuesday, February 5, 2008 - 1:20:02 PM
Last modification on : Thursday, January 28, 2021 - 10:26:02 AM
Long-term archiving on: : Thursday, September 27, 2012 - 5:49:26 PM

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  • HAL Id : tel-00239293, version 1

Citation

Stefan Le Coz. Existence, stabilité et instabilité d'ondes stationnaires pour quelques équations de Klein-Gordon et Schrödinger non linéaires. Mathématiques [math]. Université de Franche-Comté, 2007. Français. ⟨tel-00239293⟩

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