Localized Modes Due to Defects in High Contrast Periodic Media Via Two-Scale Homogenization

The spectral problem for an infinite periodic medium perturbed by a compact defect is considered. For a high contrast small ε-size periodicity and a finite size defect we consider the critical ε2-scaling for the contrast. We employ two-scale homogenization for deriving asymptotically explicit limit equations for the localized modes and associated eigenvalues. Those are expressed in terms of the eigenvalues and eigenfunctions of a perturbed version of a two-scale limit operator introduced by V. V. Zhikov with an emergent explicit nonlinear dependence on the spectral parameter for the spectral problem at the macroscale. Using the method of asymptotic expansions supplemented by a high contrast boundary layer analysis, we establish the existence of the actual eigenvalues near the eigenvalues of the limit operator, with “ε square root” error bounds. An example for circular or spherical defects in a periodic medium with isotropic homogenized properties is given

Searchlight asymptotics for high-frequency scattering by boundary inflection

We consider an inner problem for whispering gallery high-frequency asymptotic mode's scattering by a boundary inflection. The related boundary-value problem for a Schr\"{o}dinger equation on a half-line with a potential linear in both space and time appears fundamental for describing transitions from modal to scattered asymptotic patterns, and despite having been intensively studied over several decades remains largely unsolved. We prove that the solution past the inflection point has a ``searchlight'' asymptotics corresponding to a beam concentrated near the limit ray, and establish certain decay and smoothness properties of the related searchlight amplitude. We also discuss further interpretations of the above result: the existence of associated generalised wave operator, and of a version of a unitary scattering operator connecting the modal and scattered asymptotic regimes

Two-scale homogenization for a general class of high contrast PDE systems with periodic coefficients

For two-scale homogenization of a general class of asymptotically degenerating strongly elliptic symmetric PDE systems with a critically scaled high contrast periodic coefficients of a small period , we derive a two-scale limit resolvent problem under a single generic decomposition assumption for the ‘stiff’ part. We show that this key assumption does hold for a large number of examples with a high contrast, both studied before and some recent ones, including those in linear elasticity and electromagnetism. Following ideas of V. V. Zhikov, under very mild restrictions on the regularity of the domain we prove that the limit resolvent problem is well-posed and turns out to be a pseudo-resolvent problem for a well-defined non-negative self-adjoint two-scale limit operator. A key novel technical ingredient here is a proof that the linear span of product test functions in the functional spaces corresponding to the degeneracies is dense in associated two-scale energy space for a general coupling between the scales. As a result, we establish (both weak and strong) two-scale resolvent convergence, as well as some of its further implications for the spectral convergence and for convergence of parabolic and hyperbolic semigroups and of associated time-dependent initial boundary value problems. Some of the results of this work were announced in Kamotski IV, Smyshlyaev VP. Two-scale homogenization for a class of partially degenerating PDE systems. arXiv:1309.4579v1. 2013 (https://arxiv.org/abs/1309.4579v1)

Bandgaps in two-dimensional high-contrast periodic elastic beam lattice materials

We consider elastic waves in a two-dimensional periodic lattice network of Timoshenko-type beams. We show that for general configurations involving certain highly-contrasting components a high-contrast modification of the homogenization theory is capable of accounting for bandgaps, explicitly relating those to low resonant frequencies of the “soft” components. An explicit example of a square-periodic network of beams with a single isolated resonant beam within a periodicity cell is considered in detail

Contour integral solutions of the parabolic wave equation

We present a simple systematic construction and analysis of solutions of the two-dimensional parabolic wave equation that exhibit far-field localisation near certain algebraic plane curves. Our solutions are complex contour integral superpositions of elementary plane wave solutions with polynomial phase, the desired localisation being associated with the coalescence of saddle points. Our solutions provide a unified framework in which to describe some classical phenomena in two-dimensional high frequency wave propagation, including smooth and cusped caustics, whispering gallery and creeping waves, and tangent ray diffraction by a smooth boundary. We also study a subclass of solutions exhibiting localisation near a cubic parabola, and discuss their possible relevance to the study of the canonical inflection point problem governing the transition from whispering gallery waves to creeping waves

Diffraction of Creeping Waves by Conical Points

We briefly review our recent results on evaluation of the diffracted wave for electromagnetic creeping waves scattered by a conical point at a perfectly conducting surface. The theory uses matched asymptotic expansions and the reciprocity principle, and reduces the problem to the need to evaluate the "canonical" conical diffraction coefficients at the boundary. The latter is a special case of the theory developed and implemented by us before. Additional technical difficulty comes from the need to evaluate the diffraction coefficients at the boundary which within the application of our "spherical" boundary integral equation method leads to the need to evaluate appropriate singular integrals. The latter was resolved using the Discrete Fourier Transform and the whole strategy has been implemented numerically. We report sample numerical results demonstrating convergence of the algorithm