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Étude de quelques perturbations d'équations riches en symétries : résonances et stabilités

Abstract : This manuscript deals with many problems about resonance and stability. First, we design and analyse numerical methods for academic problems like the Dirichlet problem on a segment line or the transport equation associated with a two dimensional rotation. Then, we extend the classical linear analysis of Vlasov-Poisson equations near homogeneous equilibria to describe nonlinear and multidimensional phenomena. Finally, a large part of this thesis is devoted to nonlinear Schrödinger equations in dimension 1. On the one hand, we study the impact of a natural semi-discretisation on the solitary traveling waves and on the growth of the high order Sobolev norms. On the other hand, we develop a new family of normal forms to describe the dynamic of small and smooth solutions for very long times.
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Joackim Bernier. Étude de quelques perturbations d'équations riches en symétries : résonances et stabilités. Equations aux dérivées partielles [math.AP]. Université Rennes 1, 2019. Français. ⟨NNT : 2019REN1S039⟩. ⟨tel-02397827⟩

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