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

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.Comment: 48 pages, 15 figure

Epizootic Situation at Rabies in Saratov Region

The purpose of the study was to investigate epizootic situation at rabies in Saratov region during the period from 2000 to 2010. Materials and methods. Retrospective analysis of rabies morbidity of wild and farm animals was conducted. Results. During the reporting period an increase of animals' morbidity, accompanied by extension of unfavorable settlements in case of this disease was marked. Annual rate of diseased animals is defined by seasonal activity changes of foxes. The main carriers of rabies virus in the region are foxes. Conclusion. Epizootic situation at rabies in Saratov region is evaluated as unfavorabl