Multivalued Solutions to the Eikonal Equation in Stratified Media

In the present paper we study the geometric properties of the multivalued solutions to the eikonal equation and we give the appropriate classification theorems. Our motivation stems from geometrical optics for approximating high frequency waves in stratified media. We consider the case of a fixed Hamiltonian imposed by the medium, and we present the geometric framework that describes the geometric solutions, using the notion of Legendrian immersions with an initial point source or an initial smooth front. Then, we study the singularities of the solutions in the case of a smooth or piecewise Hamiltonian in a boundaryless stratified medium. Finally, we study the singularities of the solutions in a domain with a boundary that describes the propagating field in a waveguide.