“The complex eikonal equation in 2D”
Martedì 1 Giugno 2021, ore 17:00 - Zoom - Rolando Magnanini (Università degli Studi di Firenze)
Theories of monochromatic high-frequency electromagnetic fields have been designed by Keller, Felsen, Kravtsov, Ludwig, and others to represent features that are ignored by geometrical optics. These theories make use of eikonals that encode information on both phase and amplitude – in other words, they are complex-valued. Any (real-valued) geometric optical eikonal, which describes conventional rays in some light region, can be consistently continued in the shadow region beyond the relevant caustic, provided an alternative eikonal, endowed with a non-zero imaginary part, is used. In this talk, I will give an account of these ideas in dimension 2. In physical terms, the problem in hand amounts to detecting waves that rise beside, but on the dark side of, a given caustic. In mathematical terms, this theory entails a sistem of two first order PDEs, which decouples into two conjugate degenerate elliptic PDEs. We benefit from using a number of technical devices: hodograph transforms, artificial viscosity, functionals of the Calculus of Variations, and a Bäcklund transformation.