PhD Position F/M Wave propagation in unbounded hyperbolic media

The Inria Saclay-Île-de-France Research Centre was established in 2008. It has developed as part of the Saclay site in partnership with Paris-Saclay University and with the Institut Polytechnique de Paris .

The centre has 40 project teams , 32 of which operate jointly with Paris-Saclay University and the Institut Polytechnique de Paris; Its activities occupy over 600 people, scientists and research and innovation support staff, including 44 different nationalities.

Hyperbolic metamaterials are artificially engineered anisotropic materials which exhibit some unusual properties, such as negative refraction and backward wave propagation.
The name 'hyperbolic' comes from the respective dispersive curves (which relate the frequency and the wave vector of the plane waves propagating in such media): these curves take a form of hyperbolae, rather than circles or ellipses. This property enables them to support, for a fixed wavelength, an arbitrary large wavenumber. Their applications include enhanced particle absorption, emission, and collection, e.g. for sensors and antennas; super-resolution imaging; stealth technologies; rogue wave generation etc.
Unlike isotropic metamaterials, media with hyperbolic dispersion exist in nature, examples including crystals of hexagonal boron nitride, bismuth telluride, or even cold plasma.


From the mathematical point of view, the main particularity of the corresponding models lies in the fact that in the frequency domain the respective problem is (think of the wave equation where the time is replaced by a spatial variable), at least for a range of frequencies. This is strikingly different from classical frequency-domain problems, which are \textbf{elliptic} (think of the Laplace equation). Despite the abundance of the physics literature on this subject, to our knowledge, there exist very few works on the mathematical justification of the hyperbolic metamaterial models. An important related work is a very recent theoretical paper on the Poincar\'e problem (see Dyatlov et al. 2023)

The hyperbolicity of the PDE has serious implications on the well-posedness of underlying boundary-value problems. For example, it is well-known that the Dirichlet problem for the wave equation is in general ill-posed in bounded domains (cf. the classical article by F. John 1943). However, not much is known for the Klein-Gordon equation (wave equation with zero-order terms), especially when considered in the following settings: 1) unbounded domains (e.g. hyperbolic metamaterial waveguides), except for the free space case; 2) transmission problems, where the hyperbolic media interacts with vacuum, either in bounded or unbounded domains.

We are looking for a candidate with a strong background in numerical analysis for wave propagation, solid complex analysis skills and some knowledge of programming (MATLAB or C++).

• Subsidized meals
• Partial reimbursement of public transport costs
• Leave: 7 weeks of annual leave + 10 extra days off due to RTT (statutory reduction in working hours) + possibility of exceptional leave (sick children, moving home, etc.)
• Possibility of teleworking and flexible organization of working hours
• Professional equipment available (videoconferencing, loan of computer equipment, etc.)
• Social, cultural and sports events and activities
• Access to vocational training

Gross salary : 2.100 euros / month