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.

Effects of variation in the palm stearin: palm olein ratio on the crystallisation of a low-trans shortening

Changes in the theological properties during crystallisation and in the crystal size and morphology of blends containing rapeseed oil with varying percentages of palm stearin (POs) and palm olein (POf) have been studied. The crystals formed from all three blends were studied by confocal laser scanning microscopy, light microscopy and environmental scanning electron microscopy, which revealed the development of clusters of 3-5 individual elementary "spherulites" in the early stages of crystallisation. The saturated triacylglycerol content of the solid crystals separated at the onset of crystallisation was much greater than that in the total fat. Fat blends with a higher content of palm stearin had a more rapid nucleation rate when observed by light microscopy, and this caused an earlier change in the rheological properties of the fat during crystallisation. Using a low torque amplitude (0.005 Pa, which was within the linear viscoelastic region of all samples studied) and a frequency of 1 Hz, the viscoelastic properties of melted fat during cooling were studied. All samples, prior to crystallisation, showed weak viscoelastic liquid behaviour (G '', loss modulus >G', storage modulus). After crystallisation a more "solid like" behaviour was observed (G' similar to or greater than G ''). The blend having the highest concentration of POs was found to have the earliest onset of crystallisation (27% w/w POs; 12 mins, 22% w/w POs; 13.5 mins, 17% w/w POs, 15 mins, respectively). However, there were no significant differences in the time to the point when G' became greater than G' among the three blends. (c) 2006 Elsevier Ltd. All rights reserved.

Effects of composition on fat rheology and crystallisation

The crystallisation behaviour of three fat blends, comprising a commercial shortening, a blend of fats with a very low trans fatty acid content ("low-trans") and a blend including hardened rapeseed oil with a relatively high trans fatty acid content ("high-trans") was studied. Molten fats were lowered to a temperature of 31 degrees C and stirred for 0, 15, 30, 45 and 60 min. Samples were removed and their rheological properties studied, using a controlled stress rheometer, employing a frequency sweep procedure. Effects of the progressive crystallisation at 31 degrees C on the melting profile of fat samples removed from the stirred vessel and solidified at -20 degrees C were also studied by differential scanning calorimetry (DSC). The rheological profiles obtained suggested that all of the fats studied had weak viscoelastic "liquid" structures when melted, but these changed to structures perceived by the rheometer as weak viscoelastic "gels" in the early stages of crystallisation (G' (storage modulus) > G" (loss modulus) over most of the measured frequency range). These subsequently developed into weak viscoelastic semi-solids, showing frequency dependent behaviour on further crystallisation. These changes in behaviour were interpreted as changes from a small number of larger crystals "cross-linking" in a liquid matrix to a larger number of smaller crystals packed with a "slip plane" of liquid oil between them. The rate of crystallisation of the three fats was in the order high trans > low-trans > commercial shortening. Changes in the DSC melting profile due to fractionation of triacylglycerols during the crystallisation at 31 degrees C were evident for all three fats. (c) 2006 Elsevier Ltd. All rights reserved.

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.

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.

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.