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.

Interaction and Persuasion:An analysis of the use of rhetorical devices in Gordon Brown's speech to the Labour Party Conference, on September 25, 2006

This essay has identified and analysed rhetorical devices in Gordon Brown’s speech delivered at the Labour Party conference on September 25, 2006. The aim of the study was to identify specific rhetorical devices which are described as interactional resources, analyse their uses and discuss possible effects that they may have when included in a political speech. The results are based on my own interpretations but are supported by information provided in current literature by analysts and researchers of rhetoric use. The result findings could probably serve as evidence of the need for better understanding of the devices used by politicians in their relentless endeavours to influence audience decisions.

Klein-Gordon particles in mixed vector-scalar inversely linear potentials

The problem of a spinless particle subject to a general mixing of vector and scalar inversely linear potentials in a two-dimensional world is analyzed. Exact bounded solutions are found in closed form by imposing boundary conditions on the eigenfunctions which ensure that the effective Hamiltonian is Hermitian for all the points of the space. The nonrelativistic limit of our results adds a new support to the conclusion that even-parity solutions to the nonrelativistic one-dimensional hydrogen atom do not exist. (c) 2005 Elsevier B.V. All rights reserved.

A NEW HOST RECORD, PALAEMONETES CARTERI (GORDON, 1935) (DECAPODA, PALAEMONIDAE), FOR TELOTHA HENSELII (VON MARTENS, 1869) (ISOPODA, CYMOTHOIDAE)

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Equivalence of many-photon Green's functions in the Duffin-Kemmer-Petiau and Klein-Gordon-Fock statistical quantum field theories

Using the functional integral formalism for the statistical generating functional in the statistical (finite temperature) quantum field theory, we prove the equivalence of many-photon Greens functions in the Duffin-Kennner-Petiau and Klein-Gordon-Fock statistical quantum field theories. As an illustration, we calculate the one-loop polarization operators in both theories and demonstrate their coincidence.

Conformal Klein-Gordon equations and quasinormal modes

Using conformal coordinates associated with conformal relativity-associated with de Sitter spacetime homeomorphic projection into Minkowski spacetime-we obtain a conformal Klein-Gordon partial differential equation, which is intimately related to the production of quasi-normal modes (QNMs) oscillations, in the context of electromagnetic and/or gravitational perturbations around, e.g., black holes. While QNMs arise as the solution of a wave-like equation with a Poschl-Teller potential, here we deduce and analytically solve a conformal 'radial' d'Alembert-like equation, from which we derive QNMs formal solutions, in a proposed alternative to more completely describe QNMs. As a by-product we show that this 'radial' equation can be identified with a Schrodinger-like equation in which the potential is exactly the second Poschl-Teller potential, and it can shed some new light on the investigations concerning QNMs.

Camassa-Holm equation: transformation to deformed sinh-Gordon equations, cuspon and soliton solutions

In this paper a relation between the Camassa-Holm equation and the non-local deformations of the sinh-Gordon equation is used to study some properties of the former equation. We will show that cuspon and soliton solutions can be obtained from soliton solutions of the deformed sinh-Gordon equation.

The complex sine-Gordon equation as a symmetry flow of the AKNS hierarchy

It is shown how the complex sine-Gordon equation arises as a symmetry flow of the AKNS hierarchy. The AKNS hierarchy is extended by the 'negative' symmetry flows forming the Borel loop algebra. The complex sine-Gordon and the vector nonlinear Schrodinger equations appear as lowest-negative and second-positive flows within the extended hierarchy. This is fully analogous to the well known connection between the sine-Gordon and mKdV equations within the extended mKdV hierarchy. A general formalism for a Toda-like symmetry occupying the 'negative' sector of the sl(N) constrained KP hierarchy and giving rise to the negative Borel sl(N) loop algebra is indicated.