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Emphasising nonlinear behaviors for cubic coupled Schrödinger systems

Abstract : The aim of this work is to propose a study of various nonlinear behaviors for a system of two coupled cubic Schrödinger equations. Depending on the choice of the spatial domain, we highlight different examples of nonlinear behaviors. In the first chapter, we introduce the notions and tools needed to understand the issue. In particular, we justify this choice of model by recent results on the nonlinear Schrödinger equation. The second chapter is dedicated to the study of this system on the torus (one periodic coordinate). Here, we exhibit an energy exchange for long (but finite) times between different Fourier modes of the solutions: this is the (possibly shifted) beating effect. The third chapter deals with the study of the system on the real line (one Euclidean coordinate). We set up a modified scattering result to get a nonlinear behavior in infinite time. Finally, in the fourth chapter, we consider a product space (one Euclidean coordinate and one periodic coordinate). We obtain the main result of this thesis: an energy exchange in infinite time thanks to a modified scattering result.
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https://hal.archives-ouvertes.fr/tel-01562293
Contributor : Victor Vilaça Da Rocha Connect in order to contact the contributor
Submitted on : Wednesday, July 19, 2017 - 4:09:01 PM
Last modification on : Wednesday, April 27, 2022 - 4:23:51 AM

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  • HAL Id : tel-01562293, version 2

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Victor Vilaça da Rocha. Emphasising nonlinear behaviors for cubic coupled Schrödinger systems. Analysis of PDEs [math.AP]. Université de Nantes Faculté des sciences et des techniques, 2017. English. ⟨tel-01562293v2⟩

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