A universal magnification theorem for higher-order caustic singularities

Abstract

We prove that, independent of the choice of a lens model, the total signed magnification always sums to zero for a source anywhere in the four-image region close to swallowtail, elliptic umbilic, and hyperbolic umbilic caustics. This is a more global and higher-order analog of the well-known fold and cusp magnification relations, in which the total signed magnification in the two-image region of the fold, and the three-image region of the cusp, are both always zero. As an application, we construct a lensing observable for the hyperbolic umbilic magnification relation and compare it with the corresponding observables for the cusp and fold relations using a singular isothermal ellipsoid lens. We demonstrate the greater generality of the hyperbolic umbilic magnification relation by showing how it applies to the fold image doublets and cusp image triplets, and extends to image configurations that are neither. We show that the results are applicable to the study of substructure on galactic scales using observed quadruple images of lensed quasars. The magnification relations are also proved for generic 1-parameter families of mappings between planes, extending their potential range of applicability beyond lensing.

Cusps and cuspidal edges at fluid interfaces: Existence and application

“…Thus, the key distinction between the constructed solutions (12) in two dimensions and (23) in three dimensions is that the former admits a singular interface shape with nonsingular surface tension, velocity, and pressure fields. This entails the difference in the flow patterns between the 2D and 3D cases; cf.…”

“…1 -constructed here in both two and three dimensions thus establishing a relation between real physical interfaces and singularity (catastrophe) theory [22], Cusps differ from cone singularities as the angle at their apex vanishes, and are known to play an important role in many other areas of physics, e.g., gravitational lensing [23], cuspy halo in cosmology [24], and day-side cusps in magnetosphere [25], to mention a few.…”

“…While singular solutions are known in the dynamics of viscous flows, especially in fixed geometries such as the Jeffrey-Hamel flow in a converging channel [14] and on a polygon [ 15], in problems with free interfaces primarily corner [14,16] and cone [5] type solutions were studied, e.g., in the context of Taylor cones [17,18] and chemical-reaction driven tip streaming [19], The present work focuses on two generic, according to Whitney's theory [20], types of interfacial singularities [21] -cusps and cuspidal edges shown in Fig. 1 -constructed here in both two and three dimensions thus establishing a relation between real physical interfaces and singularity (catastrophe) theory [22], Cusps differ from cone singularities as the angle at their apex vanishes, and are known to play an important role in many other areas of physics, e.g., gravitational lensing [23], cuspy halo in cosmology [24], and day-side cusps in magnetosphere [25], to mention a few.In fluid dynamics, approximations [26] to cusps in the framework of macroscopic (continuum) theory were studied before in two dimensions with the methods of complex variable theory by Jeong and Moffatt [27] in the case of clean interface and by Antanovskii [6] in the presence of surfactants; however, due to the requirement of analyticity of a conformal mapping, these studies were limited to regular solutions, i.e., when the interfacial curvature remains finite except for the case of surface tension vanishing everywhere [27], not just locally. In contrast, the second key question in the present work is on the necessary condition for the existence of the genuine cusp and cuspidal edge singularities, which, as will be shown here, is a variation of surface tension thus bringing Marangoni phenomena [28]-fluid flows result ing from variations of interfacial tension-into the picture.…”

“…While singular solutions are known in the dynamics of viscous flows, especially in fixed geometries such as the Jeffrey-Hamel flow in a converging channel [14] and on a polygon [ 15], in problems with free interfaces primarily corner [14,16] and cone [5] type solutions were studied, e.g., in the context of Taylor cones [17,18] and chemical-reaction driven tip streaming [19], The present work focuses on two generic, according to Whitney's theory [20], types of interfacial singularities [21] -cusps and cuspidal edges shown in Fig. 1 -constructed here in both two and three dimensions thus establishing a relation between real physical interfaces and singularity (catastrophe) theory [22], Cusps differ from cone singularities as the angle at their apex vanishes, and are known to play an important role in many other areas of physics, e.g., gravitational lensing [23], cuspy halo in cosmology [24], and day-side cusps in magnetosphere [25], to mention a few.…”

Cusp summations and cusp relations of simple quad lenses

“…If the source infinitely approaches the fold line, the fold relation R fold will also be close to 0. In some previous works, when the point source infinitely approaches the cusp or the fold, the numerators in Equations (1) and (2) are also considered to be equal to 0 (Zakharov 1995;Aazami & Petters 2009;Petters & Werner 2010). In our recent work, we (Chu, Lin & Yang 2013) proved that Scusp and S fold are usually not equal to 0.…”