The Dirichlet-to-Neumann map for the elliptic sine Gordon

We analyse the Dirichlet problem for the elliptic sine Gordon equation in the upper half plane. We express the solution $q(x,y)$ in terms of a Riemann-Hilbert problem whose jump matrix is uniquely defined by a certain function $b(\la)$, $\la\in\R$, explicitly expressed in terms of the given Dirichlet data $g_0(x)=q(x,0)$ and the unknown Neumann boundary value $g_1(x)=q_y(x,0)$, where $g_0(x)$ and $g_1(x)$ are related via the global relation $\{b(\la)=0$, $\la\geq 0\}$. Furthermore, we show that the latter relation can be used to characterise the Dirichlet to Neumann map, i.e. to express $g_1(x)$ in terms of $g_0(x)$. It appears that this provides the first case that such a map is explicitly characterised for a nonlinear integrable {\em elliptic} PDE, as opposed to an {\em evolution} PDE.

The child and the letter: Anne Brontë’s The Tenant of Wildfell Hall

This essay engages with the question of childhood in Anne Brontë’s The Tenant of Wildfell Hall. Despite narrating a conflict concerning child custody, childhood is a subject rarely broached by the critics of the text. Indeed, the only instance of the child being addressed in criticism grants it the power to enclose potentially subversive narrative. This is a function attributed to the framing structure of the novel by other critics. This essay returns the child to Brontë’s text as a disruptive rather than containing force. Through a detailed close analysis of the novel I track such disruptions, and the extent to which these point to wider theoretical or methodological difficulties in critical accounts of childhood and absence in literature. The essay makes interventions into psychoanalysis, childhood studies, and the discourse of ‘framing’ within C19th literature.

Boundary value problems for the elliptic sine-Gordon equation in a semi-strip

We study boundary value problems posed in a semistrip for the elliptic sine-Gordon equation, which is the paradigm of an elliptic integrable PDE in two variables. We use the method introduced by one of the authors, which provides a substantial generalization of the inverse scattering transform and can be used for the analysis of boundary as opposed to initial-value problems. We first express the solution in terms of a 2 by 2 matrix Riemann-Hilbert problem whose \jump matrix" depends on both the Dirichlet and the Neumann boundary values. For a well posed problem one of these boundary values is an unknown function. This unknown function is characterised in terms of the so-called global relation, but in general this characterisation is nonlinear. We then concentrate on the case that the prescribed boundary conditions are zero along the unbounded sides of a semistrip and constant along the bounded side. This corresponds to a case of the so-called linearisable boundary conditions, however a major difficulty for this problem is the existence of non-integrable singularities of the function q_y at the two corners of the semistrip; these singularities are generated by the discontinuities of the boundary condition at these corners. Motivated by the recent solution of the analogous problem for the modified Helmholtz equation, we introduce an appropriate regularisation which overcomes this difficulty. Furthermore, by mapping the basic Riemann-Hilbert problem to an equivalent modified Riemann-Hilbert problem, we show that the solution can be expressed in terms of a 2 by 2 matrix Riemann-Hilbert problem whose jump matrix depends explicitly on the width of the semistrip L, on the constant value d of the solution along the bounded side, and on the residues at the given poles of a certain spectral function denoted by h. The determination of the function h remains open.

Understanding and representing atmospheric convection across scales: recommendations from the meeting held at Dartington Hall, Devon, UK, 28–30 January 2013

As weather and climate models move toward higher resolution, there is growing excitement about potential future improvements in the understanding and prediction of atmospheric convection and its interaction with larger-scale phenomena. A meeting in January 2013 in Dartington, Devon was convened to address the best way to maximise these improvements, specifically in a UK context but with international relevance. Specific recommendations included increased convective-scale observations, high-resolution virtual laboratories, and a system of parameterization test beds with a range of complexities. The main recommendation was to facilitate the development of physically based convective parameterizations that are scale-aware, non-local, non-equilibrium, and stochastic.

The elliptic sine-Gordon equation: a nonlinear elliptic integrable PDE

In this paper, we summarise this recent progress to underline the features specific to this nonlinear elliptic case, and we give a new classification of boundary conditions on the semistrip that satisfy a necessary condition for yielding a boundary value problem can be effectively linearised. This classification is based on formulation the equation in terms of an alternative Lax pair.