On complex-valued 2D eikonals. Part four: continuation past a caustic
| dc.creator | Magnanini, Rolando | |
| dc.creator | Talenti, Giorgio | |
| dc.date | 2009-05-16 | |
| dc.date.accessioned | 2026-07-07T13:15:54Z | |
| dc.date.available | 2026-07-07T13:15:54Z | |
| dc.description | 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. | |
| dc.description | 48 pages, 15 figures | |
| dc.identifier | https://arxiv.org/abs/0905.2690 | |
| dc.identifier | http://arxiv.org/abs/0905.2690 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/230624 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | 35F25; 35Q60; 78A05; 65D25 | |
| dc.title | On complex-valued 2D eikonals. Part four: continuation past a caustic | |
| dc.type | text |