Parametric resonance is a particular type of resonance in which a parameter in a system changes with time. A particularly interesting case is when the parameter changes in a periodic way, which can lead to very intricate behavior. This di↵ers from periodic forcing in that solutions are not necessarily periodic. A system in which parametric resonance is realized is when a fluid bath is shaken periodically, which leads to an e↵ective time dependent gravitational force. This system will be used to study the onset of surface waves in a bath with non-uniform topography. A linear model for the surface waves is derived from the Euler equations in the limit of shallow waves, which includes the geometry of the bottom and surface tension. Experiments are performed to compare with the proposed model and good qualitative agreement is found. Another experiment which relies on a shaking fluid bath is that of bouncing fluid droplets. In the case of two droplets the shaking allows for a larger bouncing droplet to attract a smaller moving droplet in a way that creates a bound system. This bound system is studied and shows some analogous properties to quantum systems, so a quantum mechanical model for a two dimensional atom is studied, as well as a proposed model for the droplet-wave system in terms of equations of fluid mechanics.

Description of the interaction of a shallow-water wave with variable topography is a classical and fundamental problem of fluid mechanics. The behaviour of linear waves and isolated solitary waves propagating over an uneven bottom is well understood. Much less is known about the propagation of nonlinear wavetrains over obstacles. For shallow-water waves, the nonlinear wavetrains are often generated in the form of undular bores, connecting two different basic flow states and having the structure of a slowly modulated periodic wave with a solitary wave at the leading edge. In this thesis, we examine the propagation of shallow-water undular bores over a nonuniform environment, and also subject to the effect of weak dissipation (turbulent bottom friction or volume viscosity). The study is performed in the framework of the variable-coefficient Korteweg-de Vries (vKdV) and variable-coefficient perturbed Korteweg-de Vries (vpKdV) equations. The behaviour of undular bores is compared with that of isolated solitary waves subject to the same external effects. We show that the interaction of the undular bore with variable topography can result in a number of adiabatic and non-adiabatic effects observed in different combinations depending on the specific bottom profile. The effects include: (i) the generation of a sequence of isolated solitons -- an expanding large-amplitude modulated solitary wavetrain propagating ahead of the bore; (ii) the generation of an extended weakly nonlinear wavetrain behind the bore; (iii) the formation of a transient multi-phase region inside the bore; (iv) a nonlocal variation of the leading solitary wave amplitude; (v) the change of the characteristics wavelength in the bore; and (vi) occurrence of a ``modulation phase shift" due to the interaction. The non-adiabatic effects (i) -- (iii) are new and to the best of our knowledge, have not been reported in previous studies. We use a combination of nonlinear modulation theory and numerical simulations to analyse these effects. In our work, we consider four prototypical variable topography profiles in our study: a slowly decreasing depth, a slowly increasing depth , a smooth bump and a smooth hole, which leads to qualitatively different undular bore deformation depending on the geometry of the slope. Also, we consider (numerically) a rapidly varying depth topography, a counterpart of the ``soliton fission" configuration. We show that all the effects mentioned above can also be observed when the undular bore propagates over a rapidly changing bottom . We then consider the modification of the variable topography effects on the undular bore by considering weak dissipation due to turbulent bottom friction or volume viscosity. The dissipation is modelled by appropriate right-hand side terms in the vKdV equation. The developed methods and results of our work can be extended to other problems involving the propagation of undular bores (dispersive shock waves in general) in variable media.

Cette thèse est dédiée à la modélisation et à l'analyse mathématique de modèles asymptotiques utilisés en océanographie décrivant la propagation des ondes internes à l'interface entre deux couches de fluides de densités différentes, soumis à la seule force de gravité.L'objectif de cette thèse est de construire et justifier de nouveaux modèles asymptotiques prenant en compte la variation de la topographie. Pour ce faire, on pose plusieurs hypothèses de petitesse sur la profondeur de l'eau et sur les déformations à l'interface et au fond. On s'intéresse plus particulièrement à deux régimes de variations topographiques, celui de moyenne amplitude et celui de lentes variations de grande amplitude.La première partie de cette thèse consiste à justifier rigoureusement et étudier mathématiquement (existence, unicité, stabilité et convergence de la solution) deux classes de modèles asymptotiques. Une classe de modèles couplés et une classe de modèles scalaires. Cette dernière classe est caractérisée par la description de la propagation unidirectionnelle des ondes internes.Dans la deuxième partie on propose un schéma numérique pour résoudre le modèle asymptotique couplé dérivé dans la première partie dans le cadre d'un font plat. Ce modèle existant dans la littérature a été reformulé d'une façon plus appropriée pour la résolution numérique en gardant le même ordre de précision que l'original et en améliorant ses propriétés de dispersion. Enfin nous présentons plusieurs simulations numériques pour valider notre schéma. / This thesis is dedicated to the modeling and the mathematical analysis of asymptotic models used in oceanography describing the propagation of internal waves at the interface between two layers of fluids of different densities, under the only influence of gravity.We aim here at constructing and justifying new asymptotic models taking into account variable topography. To this end, we assume several smallness assumptions on the depth of the water and on the deformations at the interface and at the bottom. We are interested in two topographic regimes, one for variations of medium amplitude and one for slow variations with large amplitude.In the first part of this thesis we rigorously justify and mathematically study (existence, uniqueness, stability and convergence of the solution) two classes of asymptotic models. A class of coupled models and a class of scalar models. The latter class is characterized by the description of the propagation of unidirectional internal waves. In the second part we propose a numerical resolution for the coupled asymptotic model derived in the first part restricted to the flat bottom case. This existing model in the literature has been rewritten under a new formulation more suitable for numericalresolution with the same order of precision as the standard one but with improved frequency dispersion. Finally, we present several numerical simulations to validate our scheme.