Related papers: Diffractive Propagation on Conic Manifolds

In this paper, we investigate the geometric propagation and diffraction of singularities of solutions to the wave equation on manifolds with edge singularities.

Let $(X,g)$ be a compact manifold with conic singularities. Taking $\Delta_g$ to be the Friedrichs extension of the Laplace-Beltrami operator, we examine the singularities of the trace of the half-wave group $e^{- i t \sqrt{…

We introduce a general framework for the study of the diffraction of waves by cone points at high frequencies. We prove that semiclassical regularity propagates through cone points with an almost sharp loss even when the underlying operator…

For the wave equation associated to the Laplacian on a compact manifold with boundary with a conic metric (with respect to which the boundary is metrically a point) the propagation of singularities through the boundary is analyzed. Under…

We investigate the singularities of the trace of the half-wave group, $\mathrm{Tr} \, e^{-it\sqrt\Delta}$, on Euclidean surfaces with conical singularities $(X,g)$. We compute the leading-order singularity associated to periodic orbits with…

We analyze the propagation properties of the numerical versions of one and two-dimensional wave equations, semi-discretized in space by finite difference schemes. We focus on high-frequency solutions whose propagation can be described, both…

Let $X$ be a manifold with boundary, endowed with a metric with conic singularities at the boundary components of $X$. Let $u$ be a solution to the wave equation on $\mathbb{R} \times X$. When a singularity of $u$ strikes a cone point of…

We consider manifolds with conic singularites that are isometric to $\mathbb{R}^{n}$ outside a compact set. Under natural geometric assumptions on the cone points, we prove the existence of a logarithmic resonance-free region for the…

We consider the fundamental solution to the wave equation on a manifold with corners of arbitrary codimension. If the initial pole of the solution is appropriately situated, we show that the singularities which are diffracted by the corners…

In this paper we describe the propagation of smooth (C^\infty) and Sobolev singularities for the wave equation on smooth manifolds with corners M equipped with a Riemannian metric g. That is, for X=MxR, P=D_t^2-\Delta_M, and u locally in…

We consider a nonlinear Schroedinger equation in two spatial dimensions subject to a periodic honeycomb lattice potential. Using a multi-scale expansion together with rigorous error estimates, we derive an effective model of nonlinear Dirac…

In this paper we consider a singular wave equation with distributional and more singular non-distributional coefficients and develop tools and techniques for the phase-space analysis of such problems. In particular we provide a detailed…

In this paper, we study the propagation of singularities (in the sense of $\mathcal{C}^{\infty}$ wave front set) of the solution of a model case initial-boundary value problem with glancing rays for a concave domain on an asymptotically…

For a large class of complete, non-compact Riemannian manifolds, $(M,g)$, with boundary, we prove high energy resolvent estimates in the case where there is one trapped hyperbolic geodesic. As an application, we have the following local…

For the $1+1$ dimensional damped stochastic Klein-Gordon equation, we show that random singularities associated with the law of the iterated logarithm exist and propogate in the same way as the stochastic wave equation. This provides…

H{\"o}rmander's propagation of singularities theorem does not fully describe the propagation of singularities in subelliptic wave equations, due to the existence of doubly characteristic points. In the present work, building upon a…

We investigate the diffraction of singularities of solutions to the linear elastic equation on manifolds with edge singularities. Such manifolds are modeled on the product of a smooth manifold and a cone over a compact fiber. For the…

In this paper we consider an acoustic problem of wave propagation through a discontinuous medium. The problem is reduced to the dissipative wave equation with distributional dissipation. We show that this problem has a so-called very weak…

We study the propagation of surface waves across structured surfaces with random, localized inhomogeneities. A discrete analogue of Gurtin-Murdoch model is employed and surface elasticity, in contrast to bulk elasticity, is captured by…

Linear and nonlinear mechanisms for conical wave propagation in two-dimensional lattices are explored in the realm of phononic crystals. As a prototypical example, a statically compressed granular lattice of spherical particles arranged in…