Bifurcations locales et instabilités dans des modèles issus de l'optique et de la mécanique des fluides

Abstract : In this thesis we present several contributions to qualitative study of solutions of nonlinear partial differential equations in optics and fluid mechanics models. More precisely, we focus on the existence of solutions and their stability properties. In Chapter 1, we study the Lugiato-lefever equation, which is a variant of the nonlinear Schrödinger equation arising in sereval contexts in nonlinear optics. Using tools from bifurcation and normal forms theory, we perfom a systematic analysis of stationary solutions of this equation and prove the existence of periodic and localized solutions. In Chapter 2, we present a simple criterion for linear instability of nonlinear waves. We then apply this result to the Lugiato-Lefever equation, to the Kadomtsev-Petviashvili-I equation and the Davey-Stewartson equations. These last two equations are model equations arising in fluid mechanics. In Chapter 3, we prove a criterion for linear instability of periodic solutions with small amplitude, with respect to certain quasiperiodic perturbations. This result is then applied to the Lugiato-Lefever equation.
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Cyril Godey. Bifurcations locales et instabilités dans des modèles issus de l'optique et de la mécanique des fluides. Equations aux dérivées partielles [math.AP]. Université Bourgogne Franche-Comté, 2017. Français. ⟨NNT : 2017UBFCD008⟩. ⟨tel-01790907⟩

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