The elliptic sine-Gordon equation in a half plane

We consider boundary value problems for the elliptic sine-Gordon equation posed in the half plane y > 0. This problem was considered in Gutshabash and Lipovskii (1994 J. Math. Sci. 68 197–201) using the classical inverse scattering transform approach. Given the limitations of this approach, the results obtained rely on a nonlinear constraint on the spectral data derived heuristically by analogy with the linearized case. We revisit the analysis of such problems using a recent generalization of the inverse scattering transform known as the Fokas method, and show that the nonlinear constraint of Gutshabash and Lipovskii (1994 J. Math. Sci. 68 197–201) is a consequence of the so-called global relation. We also show that this relation implies a stronger constraint on the spectral data, and in particular that no choice of boundary conditions can be associated with a decaying (possibly mod 2π) solution analogous to the pure soliton solutions of the usual, time-dependent sine-Gordon equation. We also briefly indicate how, in contrast to the evolutionary case, the elliptic sine-Gordon equation posed in the half plane does not admit linearisable boundary conditions.

The Klein-Gordon equation in a domain with time-dependent boundary

We solve a Dirichlet boundary value problem for the Klein–Gordon equation posed in a time-dependent domain. Our approach is based on a general transform method for solving boundary value problems for linear and integrable nonlinear PDE in two variables. Our results consist of the inversion formula for a generalized Fourier transform, and of the application of this generalized transform to the solution of the boundary value problem.

The Klein-Gordon equation on the half line: a Riemann-Hilbert approach

We solve an initial-boundary problem for the Klein-Gordon equation on the half line using the Riemann-Hilbert approach to solving linear boundary value problems advocated by Fokas. The approach we present can be also used to solve more complicated boundary value problems for this equation, such as problems posed on time-dependent domains. Furthermore, it can be extended to treat integrable nonlinearisations of the Klein-Gordon equation. In this respect, we briefly discuss how our results could motivate a novel treatment of the sine-Gordon equation.

Spectral analysis of the elliptic sine-Gordon equation in the quarter plane

We study the elliptic sine-Gordon equation in the quarter plane using a spectral transform approach. We determine the Riemann-Hilbert problem associated with well-posed boundary value problems in this domain and use it to derive a formal representation of the solution. Our analysis is based on a generalization of the usual inverse scattering transform recently introduced by Fokas for studying linear elliptic problems.

Megnin re-analysed: the case of the newborn baby girl, Paris, 1878

This paper analyses acarological evidence from a 130-year-old forensic investigation. It was the first case in forensic acarology, i.e., the first case where mites provided substantial information to estimate the post-mortem interval (PMI). In 1878, the mites found in the mummified body of a newborn baby girl in Paris, France, were studied by acarologist and forensic entomologist Jean Pierre M,gnin. M,gnin estimated around 2.4 million mites in the skull and identified them as Tyroglyphus longior (Gervais), a junior synonym of Tyrophagus longior. He suggested that the arrival of these mites at the corpse would have occurred by phoresy on carrier insects, roughly 5 months before the autopsy. There is no doubt about the identification of the mites, M,gnin was a highly respected acarologist. However, two main factors affecting the biology of Tyrophagus mites were not included in the original analysis. First, M,gnin stated that the mites were phoretic. However, he probably did not have access to information about the natural history of the species, because as a rule Tyrophagus mites are non-phoretic. Considering the omnipresence of Tyrophagus mites in soil, most likely the mites will have arrived almost immediately after death. Second, temperature was not taken into account during the estimations of the mite population growth rate. The new analysis is based on current knowledge of Tyrophagus biology and includes temperature, estimated following a handful of weather reports of the years 1877 and 1878. The new projections indicate that non-phoretic mites may have colonised the body just after death and the colony would have built up over 8 months, contrary to the 5 months proposed by M,gnin. This new lapse of time agrees with the PMI proposed by Brouardel: on 15 January 1878 he postulated the death of the newborn to have occurred some 8 months before the autopsy.

A Lady of York: migration, ethnicity and identity in Roman Britain

Modern methods of analysis applied to cemeteries have often been used in our pages to suggest generalities about mobility and diet. But these same techniques applied to a single individual, together with the grave goods and burial rite, can open a special kind of personal window on the past. Here, the authors of a multidisciplinary project use a combination of scientific techniques to illuminate Roman York, and later Roman history in general, with their image of a glamorous mixed-race woman, in touch with Africa, Christianity, Rome and Yorkshire.

The Identity of the St Bees Lady, Cumbria: An Osteobiographical Approach

Using an Osteobiographical approach, this contribution considers the identity of the woman found alongside the St Bees Man, one of the best-preserved archaeological bodies ever discovered. Osteological, isotopic and radiocarbon analyses, combined with the archaeological context of the burial and documented social history, provide the basis for the identification of a late 14th-century heiress whose activities were at the heart of medieval northern English geopolitics.