Exact wave-optical imaging of a Kerr-de Sitter black hole using Heun's equation

Spacetime perturbations due to scalar, vector, and tensor fields on a fixed background geometry can be described in the framework of Teukolsky's equation. In this work, wave scattering is treated analytically, using the Green's function method and solutions to the separated radial and angular differential equations in combination with a partial wave technique for a scalar and monochromatic perturbation. The results are applied to analytically describe wave-optical imaging via Kirchhoff-Fresnel diffraction, leading to, e.g., the formation of observable black hole shadows. A comparison to the ray-optical description is given, providing new insights into wave-optical effects and properties. On a Kerr-de Sitter spacetime, the cosmological constant changes the singularity structure of the Teukolsky equation and allows for an analytical, exact solution via a transformation into the Heun's differential equation, which is the most general, second-order differential equation with four regular singularities. The scattering of waves originating from a point source involves a solution in terms of the so-called Heun's function $Hf$. It is used to find angular solutions, which form a complete set of orthonormal functions similar to the spherical harmonics. Our approach allows to solve the scattering problem while taking into account the complex interplay of Heun's functions around local singularities.Comment: 27 pages, 15 figure

Analytical Solutions to General Anti-Plane Shear Problems In Finite Elasticity

This paper presents a pure complementary energy variational method for solving anti-plane shear problem in finite elasticity. Based on the canonical duality-triality theory developed by the author, the nonlinear/nonconex partial differential equation for the large deformation problem is converted into an algebraic equation in dual space, which can, in principle, be solved to obtain a complete set of stress solutions. Therefore, a general analytical solution form of the deformation is obtained subjected to a compatibility condition. Applications are illustrated by examples with both convex and nonconvex stored strain energies governed by quadratic-exponential and power-law material models, respectively. Results show that the nonconvex variational problem could have multiple solutions at each material point, the complementary gap function and the triality theory can be used to identify both global and local extremal solutions, while the popular (poly-, quasi-, and rank-one) convexities provide only local minimal criteria, the Legendre-Hadamard condition does not guarantee uniqueness of solutions. This paper demonstrates again that the pure complementary energy principle and the triality theory play important roles in finite deformation theory and nonconvex analysis.Comment: 23 pages, 4 figures. Mathematics and Mechanics of Solids, 201

Wave mechanics in media pinned at Bravais lattice points

The propagation of waves through microstructured media with periodically arranged inclusions has applications in many areas of physics and engineering, stretching from photonic crystals through to seismic metamaterials. In the high-frequency regime, modelling such behaviour is complicated by multiple scattering of the resulting short waves between the inclusions. Our aim is to develop an asymptotic theory for modelling systems with arbitrarily-shaped inclusions located on general Bravais lattices. We then consider the limit of point-like inclusions, the advantage being that exact solutions can be obtained using Fourier methods, and go on to derive effective medium equations using asymptotic analysis. This approach allows us to explore the underlying reasons for dynamic anisotropy, localisation of waves, and other properties typical of such systems, and in particular their dependence upon geometry. Solutions of the effective medium equations are compared with the exact solutions, shedding further light on the underlying physics. We focus on examples that exhibit dynamic anisotropy as these demonstrate the capability of the asymptotic theory to pick up detailed qualitative and quantitative features

Stochastic fiber dynamics in a spatially semi-discrete setting

We investigate a spatially discrete surrogate model for the dynamics of a slender, elastic, inextensible fiber in turbulent flows. Deduced from a continuous space-time beam model for which no solution theory is available, it consists of a high-dimensional second order stochastic differential equation in time with a nonlinear algebraic constraint and an associated Lagrange multiplier term. We establish a suitable framework for the rigorous formulation and analysis of the semi-discrete model and prove existence and uniqueness of a global strong solution. The proof is based on an explicit representation of the Lagrange multiplier and on the observation that the obtained explicit drift term in the equation satisfies a one-sided linear growth condition on the constraint manifold. The theoretical analysis is complemented by numerical studies concerning the time discretization of our model. The performance of implicit Euler-type methods can be improved when using the explicit representation of the Lagrange multiplier to compute refined initial estimates for the Newton method applied in each time step.Comment: 20 pages; typos removed, references adde