Description du sujet de thèse

Solitons are special types of waves that can spread without deformation over long distances. This particular behavior has its origin in a balance between nonlinear effects and dispersion. Solitons may appear in many different physical systems, but the best known context is probably that of water surface waves. In this case, in the shallow water limit, long wavelength perturbations obey the Korteweg-de Vries equation (KdV). Soliton solutions of the KdV equation are now well understood. However, the problem becomes much more delicate when you leave the ideal regime of the KdV equation. In particular, if the water is not stationary but flows on a submerged obstacle, many new phenomena arise, such as hydraulic jumps or undular jumps.

A subject apparently unrelated to water wave physics is that of artificial black holes. If the water flows faster than the velocity of the waves, they can be trapped in the fast-flow region, analogous to light waves being trapped in the gravitational field of a black hole. The location where the flow speed crosses the wave speed is analogous to the horizon of a black hole: the point of no return beyond which the light is trapped inside. Mathematically, the analogy is established by showing that linear waves propagate over a variable flow by obeying a wave equation in a curved space-time, as in general relativity. Although the effect of wave dispersion in these fluid analogues of black holes has been studied in depth, the effect of nonlinearities is still poorly understood.

Theoretically, a wide range of flow structure has been found, but experimentally, only a much smaller set is observed. This raises the question of the stability of these stationary solutions. In general, the mathematical demonstration of stability or instability in fluid mechanics is a very difficult problem. But in the context of the KdV equation, we have a panel of powerful tools at our disposal: linear analysis, Whitham modulation theory, inverse scattering, or direct numerical resolution. The aim of this thesis topic is to understand the stability of the different flows, with an analytical and numerical approach, and then to study the consequences for experiments of similar black holes. In a second stage, we will ask ourselves how to control instability in order to obtain flows more faithful to analogy. This can be done, for example, by the careful design of submerged obstacles, which eliminate unwanted instability.

The PhD is will be funded by the JCJC WaDiToMe ANR.

Contexte de travail

The PhD will take place in the Laboratory of Mechanics and Acoustics (LMA). It will be co-supervized by Bruno Lombard and Antonin Coutant.

Informations complémentaires

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