Abstract:
I will focus on the properties of waves in 2-dimensional honeycomb structures.
Two areas where such structures have been explored extensively are
(i) condensed matter physics (graphene) and (ii) photonics.
Many of the remarkable wave propagation properties of honeycomb structures are related to the presence of
conical singularities (``Dirac points'') in the associated dispersion surfaces.
Physical modeling (since Wallace, 1947) has centered on the tight-binding (discrete) approximation of the
underlying partial differential equation (Schroedinger's eqn).
I'll present results (joint with C.L. Fefferman) on the existence of Dirac points
for the non-relativistic Schroedinger equation with a generic honeycomb lattice potential.
We also prove that the very long time dynamics of wave-packets is governed
by an effective two-dimensional Dirac system. Finally, we discuss the propagation
of nonlinear waves in honeycomb structures. |