Abstract: Periodic media appear naturally in many areas of physics, and have drawn particular interest especially with the advent of photonic crystals. Understanding how waves propagate in such media is essential for the design of practical applications.

This talk focuses on the analysis and numerical solution of PDE models for time-harmonic wave propagation at interfaces between two-dimensional periodic half-spaces. While most existing approaches assume periodicity of the medium along the interface, we investigate the more delicate case where this assumption no longer holds. Our analysis relies on the crucial observation that the medium has a quasiperiodic structure along the interface, meaning it can be viewed as a slice of a three-dimensional periodic medium. This enables us to seek solutions to our PDE as restrictions of solutions to an augmented elliptically degenerate PDE set in three dimensions, where periodicity along the interface is recovered.

Based on joint work with Sonia Fliss and Patrick Joly, I will first show how this so-called lifting approach can be used to numerically solve the time-harmonic wave equation in junctions of periodic half-spaces. I will then present ongoing work with Michael Weinstein, where the lifting approach is used to define and study edge states in honeycomb structures perturbed along irrational edges.

About the speaker

Since January 2024, I am a postdoc in the Applied Physics and Applied Mathematics (APAM) department of Columbia University. From October 2020 to December 2023, I was a PhD student in the POEMS team at Ensta Paris. My current research interests lie in the mathematical study and the numerical simulation of time-harmonic wave propagation phenomena in heterogeneous ― periodic or quasiperiodic ― structures with defects.