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Theses

Études d’oscillations non linéaires près d’une caustique

Abstract : We underscore nonlinear phenomena when oscillations give rise to a caustic. The first part of this work considers a family of semi-linear wave equations, for which it is shown that new terms appear in the asymptotic expansion of the solution, due to the caustic crossing. This study, in $L^\infty$, emphasizes a boundary layer near the caustic, inside which the exact solution and the first profile of geometric optics have radically different behaviors. The second part of this thesis is concerned with a family of nonlinear Schrödinger equations, for which several notions of critical index appear, as for the description of the propagation outside the caustic on the one hand, and the caustic crossing on the other hand. The propagation is similar to that known by WKB method, and the caustic crossing is either the same as in the linear case, or described in terms of scattering operators. A uniform description is obtained by a generalization of the use of Lagrangian integrals.
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Rémi Carles. Études d’oscillations non linéaires près d’une caustique. Equations aux dérivées partielles [math.AP]. Université de Rennes 1, 1999. Français. ⟨NNT : 1999REN10045⟩. ⟨tel-02933740⟩

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