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On Complex-valued 2D Eikonals. Part Four: Continuation Past a Caustic

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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.

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    Rolando Magnanini & Giorgio Talenti

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Lecture held by G. Talenti in the Seminario Matematico e Fisico on February 23, 2006

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Magnanini, R., Talenti, G. On Complex-valued 2D Eikonals. Part Four: Continuation Past a Caustic. Milan J. Math. 77, 1–66 (2009). https://doi.org/10.1007/s00032-009-0103-x

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