Abstract:
The Maxwell equations govern the propagation of electromagnetic waves in matter. In many cases the material
properties do not change when an electromagnetic wave propagates through them. However, for a class of materials, the refractive changes in a nonlinear way in the presence of a
sufficiently strong electric field E. In this talk I will consider a model for a class of materials with nonlinear polarization properties. I will further consider special geometries where one can prove the existence of propagating time-periodic electromagnetic waves which are localized in directions orthogonal to the propagation direction. This problem leads to a quasilinear hyperbolic nonlinear partial differential
equation for the electric field E. Solutions with the above properties (localized, time-periodic, propagating) will be found by a variational principle. Numerical simulations will also be shown.
This is joint work with Sebastian Ohrem (KIT).