On complex-valued 2D eikonals. Part four: continuation past a caustic

TL;DR

This paper develops a mathematical framework for continuing complex-valued 2D eikonals beyond caustics, addressing physical wave phenomena near shadow regions with an innovative algorithm to handle ill-posedness.

Contribution

It introduces a new approach to extend complex eikonals past caustics in 2D, combining PDE analysis with numerical methods to address degeneracy and ill-posedness.

Findings

Effective algorithm for stable approximation of eikonals beyond caustics

Demonstration of continuation of wave solutions in shadow regions

Handling of degeneracy and ill-posedness in complex eikonal problems

Abstract

Theories of monochromatic high-frequency electromagnetic fields have been designed by Felsen, Kravtsov, Ludwig and others with a view to portraying features that are ignored by geometrical optics. These theories have recourse to eikonals that encode information on both phase and amplitude -- in other words, are complex-valued. The following mathematical principle is ultimately behind the scenes: any geometric optical eikonal, which conventional rays engender 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, comes on stage. In the present paper we explore such a principle in dimension $2.$ We investigate a partial differential system that governs the real and the imaginary parts of complex-valued two-dimensional eikonals, and an initial value problem germane to it. 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, such a problem shows two main peculiarities: on the one hand, degeneracy near the initial curve; on the other hand, ill-posedness in the sense of Hadamard. We benefit from using a number of technical devices: hodograph transforms, artificial viscosity, and a suitable discretization. Approximate differentiation and a parody of the quasi-reversibility method are also involved. We offer an algorithm that restrains instability and produces effective approximate solutions.