Dans cette thèse, nous étudions plusieurs aspects de la radiation de Hawking en présence de violations de l'invariance locale de Lorentz. Ces violations sont introduites par une modification de la relation de dispersion, devenant non-linéaire aux courtes longueurs d’onde. Les principales motivations de ces travaux ont une double origine. Il y a d’une part le développement en matière condensée de trous noirs analogues, ou l’écoulement d’un fluide est perçu comme une métrique d’espace-temps pour les ondes de perturbations et ou la radiation de Hawking pourrait être détectée expérimentalement. D’autre part, il se pourrait que des effets de gravité quantique puissent être modélisés par une modification de la relation de dispersion. En premier lieu, nous avons obtenu des caractérisations précises des conditions nécessaires au maintien de l’effet Hawking en présence de violation de l’invariance de Lorentz. De plus, nous avons étudié l’apparition d’une onde macroscopique de fréquence nulle, dans des écoulements de type trous blancs et également pour des champs massifs. Une autre partie de ce travail a consisté à analyser une instabilité engendrée par les effets dispersifs, ou la radiation de Hawking est auto-amplifiée, générant ainsi un flux sortant exponentiellement croissant dans le temps. / In this thesis, we study several features of Hawking radiation in the presence of ultraviolet Lorentz violations. These violations are implemented by a modified dispersion relation that becomes nonlinear at short wavelengths. The motivations of this work arise on the one hand from the developing field of analog gravity, where we aim at measuring the Hawking effect in fluid flows that mimic black hole space-times, and on the other hand from the possibility that quantum gravity effects might be approximately modeled by a modified dispersion relation. We develop several studies on various aspects of the prob- lem. First we obtain precise characterizations about the deviations from the Hawking result of black hole radiation, which are induced by dispersion. Second, we study the emergence, both in white hole flows or for massive fields, of a macroscopic standing wave, spontaneously produced from the Hawking effect, and known as ‘undulation’. Third, we describe in detail an instability named black hole laser, which arises in the presence of two horizons, where Hawking radiation is self-amplified and induces an exponentially growing in time emitted flux.

In 1981, Unruh suggested the possibility of simulating the dynamics of quantum fields in curved spacetimes using sound-waves propagating in moving fluids: a supersonic flow would indeed influence the dynamics of sound similarly to what happens to light when it’s dragged by the spacetime geometry in strong gravity environments. This simple yet groundbreaking observation has lead to the beginning of a whole new field of research, nowadays known as Analog Gravity. Due to their superfluid character, intrinsic quantum nature and impressive experimental tunability, Bose-Einstein condensates represent one of the most promising platforms to realize analog spacetimes, including black-hole geometries with horizons and ergoregions, as well as of time-dependent configurations relevant to cosmology. In this Thesis we go beyond the standard single-component BEC and focus on two-component mixtures of atomic condensates, possibly in the presence of a coherent coupling between the two-components: the availability of various branches of elementary excitations with different sound speed and effective mass may in fact lead to advantages in the implementation of interesting geometries and, eventually, to the exploration of a broader spectrum of physical processes. We first consider black-hole related phenomena (Hawking radiation and rotational superradiance) that have already been analysed with single-component systems, generalising the results to mixtures; we then proceed to tackle a problem (the decay from the false vacuum) which instead requires the additional degrees of freedom that only a mixture displays.