Section: Research Program

Spectral theory and modal approaches

The study of waveguides is a longstanding and major topic of the team. Concerning the selfadjoint spectral theory for open waveguides, we turned recently to the very important case of periodic media. One objective is to design periodic structures with localized perturbations to create gaps in the spectrum, containing isolating eigenvalues.

Then, we would like to go further in proving the absence of localized modes in non uniform open waveguides. An original approach has been successfully applied to the scalar problem of a waveguides junctions or bent waveguides. The challenge now is to extend these ideas to vectorial problems (for applications to electromagnetism or elastodynamics) and to junctions of periodic waveguides.

Besides, we will continue our activity on modal methods for closed waveguides. In particular, we aim at extending the enriched modal method to take into account curvature and rough boundaries.

Finally, we are developing asymptotic models for networks of thin waveguides which arise in several applications (electric networks, simulation of lung, nanophotonics...).

The study of waveguides is a longstanding and major topic of the team.

On this topic, a workshop entitled « New trends in theoretical and numerical analysis of waveguides » was co-organized by Anne-Sophie Bonnet-Ben Dhia (and Philippe Briet and Eric Soccrosi from CPT, Marseille and Michel Cristofol from I2M, Marseille) at IGESA (Porquerolles) from May 16th to 19th. This workshop is part of series of workshops organised from 2011 (in 2011 at Irmar, Rennes, in 2012 at Marseille, in 2013 at POems, Palaiseau, in 2015 at Napoli). The aim of these workshops is to bring together researchers from Mathematics, mathematical physics, theoretical physics and numerical analysis in order to encourage and stimulate the interactions between the different communities on problems associated to waveguides.