Este trabalho investiga a quebra da aproximação modulacional na interação não-linear de três ondas, a interação do tripleto de ondas. Um modo comum de descrever a interação de três ondas portadoras de alta frequência é a partir da aproximação modulacional, que assume que as amplitudes e fases são lentamente moduladas. Isto apenas é verdade quando o acoplamento entre as três ondas é fraco. Ao se analisar os tipos de dinâmica envolvidas quando o valor do acoplamento é alterado, para grandes valores de acoplamento é detectada uma transição abrupta onde a amplitude limitada do regime modulacional chega a outras regiões do espaço configuracional. Também é investigado o caso onde uma das ondas do tripleto possuir uma frequência muito menor que as outras duas. / This work investigates the breakdown of the traditional modulation approximation in the three wave nonlinear interactions, the wave triplet interaction. A common way to describe the interaction of three high-frequency carriers is by the modulacional approximation, which assume that amplitude and phases are slowly modulated. This is only accurate when the three-wave coupling is weak. When analyzing the different types of dynamics when the coupling is changed, for large values of coupling it is detected an abrupt transition where the limited amplitude of the modulacional region reaches to other regions of the configuration space. It is also investigated the case where one of the waves of the triplet has a much lower frequency than the other two.

Este trabalho investiga a quebra da aproximação modulacional na interação não-linear de três ondas, a interação do tripleto de ondas. Um modo comum de descrever a interação de três ondas portadoras de alta frequência é a partir da aproximação modulacional, que assume que as amplitudes e fases são lentamente moduladas. Isto apenas é verdade quando o acoplamento entre as três ondas é fraco. Ao se analisar os tipos de dinâmica envolvidas quando o valor do acoplamento é alterado, para grandes valores de acoplamento é detectada uma transição abrupta onde a amplitude limitada do regime modulacional chega a outras regiões do espaço configuracional. Também é investigado o caso onde uma das ondas do tripleto possuir uma frequência muito menor que as outras duas. / This work investigates the breakdown of the traditional modulation approximation in the three wave nonlinear interactions, the wave triplet interaction. A common way to describe the interaction of three high-frequency carriers is by the modulacional approximation, which assume that amplitude and phases are slowly modulated. This is only accurate when the three-wave coupling is weak. When analyzing the different types of dynamics when the coupling is changed, for large values of coupling it is detected an abrupt transition where the limited amplitude of the modulacional region reaches to other regions of the configuration space. It is also investigated the case where one of the waves of the triplet has a much lower frequency than the other two.

Este trabalho investiga a quebra da aproximação modulacional na interação não-linear de três ondas, a interação do tripleto de ondas. Um modo comum de descrever a interação de três ondas portadoras de alta frequência é a partir da aproximação modulacional, que assume que as amplitudes e fases são lentamente moduladas. Isto apenas é verdade quando o acoplamento entre as três ondas é fraco. Ao se analisar os tipos de dinâmica envolvidas quando o valor do acoplamento é alterado, para grandes valores de acoplamento é detectada uma transição abrupta onde a amplitude limitada do regime modulacional chega a outras regiões do espaço configuracional. Também é investigado o caso onde uma das ondas do tripleto possuir uma frequência muito menor que as outras duas. / This work investigates the breakdown of the traditional modulation approximation in the three wave nonlinear interactions, the wave triplet interaction. A common way to describe the interaction of three high-frequency carriers is by the modulacional approximation, which assume that amplitude and phases are slowly modulated. This is only accurate when the three-wave coupling is weak. When analyzing the different types of dynamics when the coupling is changed, for large values of coupling it is detected an abrupt transition where the limited amplitude of the modulacional region reaches to other regions of the configuration space. It is also investigated the case where one of the waves of the triplet has a much lower frequency than the other two.

Cette thèse s'inscrit dans l'étude des modèles asymptotiques aux équations des ondes de surface dans le régime modulationnel. Le problème des ondes de surface consiste à décrire le mouvement - sous l'influence de la gravitation et éventuellement de tension de surface - d'un fluide dans un domaine délimité par la surface libre du fluide et par un fond fixe. Dans l'étude de ce problème, on s'intéresse en particulier aux ondes se propageant à la surface du fluide.Dans le régime modulationnel, on considère l'évolution des ondes de surface sous forme de paquets d'ondes de faible amplitude se propageant dans une direction. Il est bien connu que la motion de l'enveloppe du paquet d'onde sur une échelle de temps d'ordre t = O(1/ϵ²), où ϵ est un petit paramètre désignant l'amplitude, est décrite approximativement par des systèmes d'équations appelés systèmes de Benney-Roskes (BR) / Davey-Stewartson (DS). Ces systèmes sont donnés par une équation de type Schrödinger cubique couplée à une équation d'ondes. L'approximation classique de BR / DS est bien établie et a été largement étudiée au cours des dernières décennies. Récemment, David Lannes a introduit une version à "dispersion exacte" de ces systèmes. Contrairement aux équations de BR / DS standard, les systèmes à dispersion exacte préservent la relation de dispersion des équations des ondes de surface. On devrait obtenir ainsi une description plus riche du vrai comportement dynamique des ondes de surface que dans le cas de l'approximation classique.Le systèmes de BR / DS à dispersion exacte sont étudiés dans cette thèse. La première partie est consacrée à la déduction formelle des systèmes de BR / DS en tant que modèles asymptotiques aux équations des ondes de surface. Nous donnons en outre un résultat sur la consistance de cette approximation.Ensuite, nous étudions le problème de Cauchy pour le système de BR à dispersion exacte. En fait, afin de justifier la consistance de l'approximation de BR avec les équations exactes, on doit prouver que ce système est bien posé (en espace de Sobolev) sur une échelle de temps d'ordre O(1/ϵ). Ceci est un problème ouvert même dans le cas classique, du moins pour le système de dimension 1 + 2. De même, nous ne pouvons pas démontrer l'existence de solutions en temps long pour le système de BR à dispersion exacte, mais nous obtenons un théorème d'existence locale (t = O(1)) à condition que la tension de surface soit assez forte. Si nous nous restreignons au système de dimension 1+1, nous pouvons enlever la contrainte sur la tension de surface. L'idée de la preuve d'existence locale, qui est inspirée par un travail de Schochet-Weinstein, est d'écrire le système de BR comme un système symétrique hyperbolique quasi-linéaire perturbé par un terme dispersif ne contribuant pas à l'énergie du système. Ainsi, nous pouvons appliquer les méthodes standard de résolution des systèmes hyperboliques.En modifiant le terme non-linéaire du système de BR de dimension 1+1 sans changer l'ordre de consistance, nous obtenons un système qui est bien posé sur l'échelle de temps appropriée O(1/ϵ). Cependant, cette démarche ne peut pas être généralisée au cas de dimension 1+2.Dans le dernier chapitre de cette thèse, nous donnons quelques résultats sur les systèmes de Davey-Stewartson à dispersion exacte. Pour les systèmes de DS, il est suffisant de démontrer qu'ils sont bien posés localement afin de justifier leur consistance avec les équations des ondes de surface. La théorie d'existence de solutions est assez complète pour le système de DS classique. Dans le cas de dispersion exacte cependant, les équations paraissent mal posées généralement, si bien que l'existence locale ne peut être démontrée pour l'instant que pour quelques cas particuliers simples. / This thesis is concerned with asymptotic models to the water wave equations in the modulational regime. The water wave equations describe the motion - under the influence of gravity and possibly surface tension - of an inviscid fluid in a domain which is bounded by a fixed bottom from below and the free surface of the fluid from above. In the study of the water wave problem, one is in particular interested by waves propagating on the surface of the fluid.In the modulational regime, one considers the evolution of surface waves under the form of small amplitude wave packets traveling in one direction. It is well known that the evolution of the wave packet envelope on the long time scale t = O(1/ϵ²), where ϵ is a small parameter denoting the amplitude of the wave, is approximately governed by a set of equations known as the Benney-Roskes (BR) / Davey-Stewartson (DS) systems. These systems are essentially given by a cubic Schrödinger-type equation coupled to a wave equation. The classical BR / DS approximation is well established and has been largely studied in the past decades. Recently, David Lannes has introduced a "full dispersion" version of these systems. In contrast to the standard BR / DS equations, the full dispersion systems preserve the linear dispersion relation of the full water wave equations, and should therefore give a richer description of the original wave dynamics than the classical approximation.The full dispersion BR / DS systems are studied in this thesis. In the first part, we formally derive the full dispersion BR / DS approximation from the water wave equations both in the case of zero and positive surface tension. The formal derivation is completed by a consistency result.We then study well-posedness in Sobolev space of the full dispersion BR system. In order to justify consistency of the BR approximation with the full water wave equations, one needs to show that the BR system is well posed on a time scale of order O(1/ϵ). This is an open problem even in the classical case, at least for the 1 + 2 dimensional system. We also do not obtain well-posedness on the long time scale for the full dispersion BR system, but we can show that it is locally well-posed in the case of sufficiently strong surface tension, and additionally in the zero surface tension case if we restrict ourselves to the 1+1 dimensional system. The proof is inspired by a paper of Schochet-Weinstein, and is based on writing the full dispersion BR system as a quasilinear symmetric hyperbolic system with dispersive perturbation, where the dispersive terms do not contribute to the energy. We can therefore apply classical solution methods for hyperbolic systems.By modifying the nonlinear part of the 1+1 dimensional full dispersion BR system without changing consistency, we obtain a system that is well-posed on the appropriate O(1/ϵ) time scale. This approach however does not generalize to the 1+2 dimensional case.In the last chapter of the thesis, we give some results on the full dispersion DS systems, which are obtained as special limits of the full dispersion BR system. For these systems, it is sufficient to prove local well-posedness in order to show consistency with the water wave equations. For the standard DS systems, local well-posedness theory is quite complete. For the full dispersion systems, the analysis is complicated by some nonlocal operators and the equations seem to be generally ill-posed. There are however some simple cases where local well-posedness can be shown. We also discuss some modifications of the full dispersion DS system that might allow to solve it for a larger range of parameters.