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

Social foraging European shags: GPS tracking reveals birds from neighbouring colonies have shared foraging grounds

Developments in tracking technologies have enhanced our understanding of the behaviours of many seabird species. However few studies have examined the social aspects of seabird foraging behaviour, despite the effect this might have on the distribution of foraging areas and the differences that might arise between colonies. Here we use bird-borne GPS and behavioural observation to study the social foraging behaviour and habitat use of breeding shags from three breeding colonies in the Isles of Scilly, UK. Thirteen breeding shags from three colonies (six at two colonies and a single bird from another) were tracked between 2010 and 2012 and related to observations of conspecific foraging aggregations (2013-2014). Tracked shags had short foraging ranges (1.74 ± 1.6 km) mostly travelling to shallow waters between the islands and observations revealed that many shags foraged in large social groups that were consistent in time and space. There were also no clear differences in foraging distributions among colonies—birds shared similar foraging grounds. Our finding provides important insight into the use of social information among foraging seabirds and how this may lead to shared foraging areas, as well as space partitioning