arXiv research

Neural networks improve wave-equation simulation accuracy.

New method finds precise late-time behavior of wave equations.

Paper explores non-uniqueness and uniqueness class for wave equations on graphs.

Study gauge freedoms in elastic wave equations and Dirichlet-to-Neumann map.

Proves wave equation solutions in Kerr-de Sitter spacetime have specific asymptotic expansions.

Solves wave equation on non-flat harmonic manifolds using Abel transform and Fourier analysis.

The paper establishes scattering theory for wave equations on Schwarzschild spacetime.

In this paper we prove the propagation of singularities for the wave equation on differential forms with natural (i.e. relative or absolute) boundary conditions on Lorentzian manifolds with corners, which in particular includes a formulation of Maxwell's equations. These results are analogous to those obtained by the a…

Study on blow-up solutions for semilinear wave equations on specific manifolds.

This is a survey on the analytic theory of linear wave equations on globally hyperbolic Lorentzian manifolds. There is no claim of originality.

In this paper, a symmetry classification of a $(2+1)$-nonlinear wave equation $u_{tt}-f(u)(u_{xx}+u_{yy})=0$ where $f(u)$ is a smooth function on $u$, using Lie group method, is given. The basic infinitesimal method for calculating symmetry groups is presented, and used to determine the general symmetry group of this $…

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

Using the scalar curvature of the product manifold S^{2}X R and the complete group classification of nonlinear Poisson equation on (pseudo) Riemannian manifolds, we extend the previous results on symmetry analysis of homogeneous wave equation obtained by H. Azad and M. T. Mustafa [H. Azad and M. T. Mustafa, Symmetry an…

In a previous paper on coupled gravitational and electromagnetic perturbations of Reissner-Nordström spacetime in a polarized setting, we derived a system of wave equations for two independent quantities, one related to the Weyl curvature and one related to the Ricci curvature of the perturbed spacetime. We analyze her…

The abstract explores a new wave equation linking quantum mechanics and complex adaptive systems.

Recovering matrix valued potentials from wave equation data on stationary spacetimes.

This paper addresses parameter estimation for wave equations with Markovian switching.

The commuting vector fields approach, devised for strichartz estimates in [13], was developed for proving the local well-posedness in the Sobolev spaces $H^s$ with $s>2+\frac{2-\sqrt{3}}{2}$ for general quasi-linear wave equation in ${\mathbb R}^{1+3}$ by Klainerman and Rodnianski. Via this approach they obtained the l…

We study the following problem: Given initial data on a compact Cauchy horizon, does there exist a unique solution to wave equations on the globally hyperbolic region? Our main results apply to any spacetime satisfying the null energy condition and containing a compact Cauchy horizon with surface gravity that can be no…

This paper surveys a few aspects of the global theory of wave equations. This material is structured around the contents of a minicourse given by the second author during the CMI/ETH Summer School on evolution equations during the Summer of 2008.

In this paper, we study the schrodinger equation and wave equation with the Dirichlet boundary condition on a connected finite graph. The explicit expressions for solutions are given and the energy conservations are derived. Applications to the corresponding nonlinear problems are indicated.

Solutions to the wave equation on de Sitter-Schwarzschild space with smooth initial data on a Cauchy surface are shown to decay exponentially to a constant at temporal infinity, with corresponding uniform decay on the appropriately compactified space.

Veronese webs are rich geometric structures with deep relationships to various domains of mathematics. The PDEs which determine the Veronese web are overdetermined if dim >3, but in the case dim =3 they reduce to a special flavor of a non-linear wave equation. The symmetries embedded in the definition of a Veronese web…

In these lecture notes we discuss the solution theory of geometric wave equations as they arise in Lorentzian geometry: for a normally hyperbolic differential operator the existence and uniqueness properties of Green functions and Green operators is discussed including a detailed treatment of the Cauchy problem on a gl…

Researchers find counterexamples to inverse problems for wave equations.

We prove that second-order hyperbolic Monge-Ampere equations for one function of two variables are connected to the wave equation by a Backlund transformation if and only if they are integrable by the method of Darboux at second order. One direction of proof, proving Darboux integrability, follows the implications of t…

In this paper we exploit the ideas and formalisms of twistor theory, to show how, on Minkowski space, given a null solution of the wave equation, there are precisely two null directions in $\ker df$, at least one of which is a shear-free ray congruence.

DistillKac generates images quickly using damped wave equations.

Study proves well-posedness and scattering for wave equations on hyperbolic spaces with singular data.

We study the Cauchy problem for the wave equation on extreme Kerr backgrounds under axisymmetry. Specifically, we consider regular axisymmetric initial data prescribed on a Cauchy hypersurface S which connects the future event horizon with spacelike or null infinity, and we solve the linear wave equation on the domain …

Geometric focusing affects dispersive estimates for Schrödinger and wave equations.

Measuring wave sources uniquely identifies manifold properties.

Carter tensor analysis aids wave equation on Kerr-Newman spacetime.

Study proves global existence and decay for complex wave equations.

Wave equation map reveals manifold's structure.

Paper approximates backward heat equation using wave equations and Ricci flow.

Study examines wave equation decay and Strichartz estimates on conic manifolds.

The moving coframe method is applied to solve the local equivalence problem for the class of nonlinear wave equations in two independent variables under an action of the pseudo-group of contact transformations. The structure equations and the complete sets of differential invariants for symmetry groups are found. The s…

Study dispersive estimates for Schrödinger and wave equations on a cone with specific metric.

Proves energy estimates for tensorial wave equations, decoupling components for stability proof.

We consider the wave equation on a closed Riemannian manifold. We observe the restriction of the solutions to a measurable subset $ω$ along a time interval $[0, T]$ with $T>0$. It is well known that, if $ω$ is open and if the pair $(ω,T)$ satisfies the Geometric Control Condition then an observability inequality is sat…

The paper explains how microlocal analysis solves geometric inverse problems.

We study the transversal wave equation on a compact Riemannian foliated manifold. As applications, we get an Egorov's type theorem for transversally elliptic operators, state a relationship between the singularities of the Fourier transform of the spectrum distribution function of a transversally elliptic operator and …

This paper concludes the series begun in [M. Dafermos and I. Rodnianski, Decay for solutions of the wave equation on Kerr exterior spacetimes I-II: the cases |a| << M or axisymmetry, arXiv:1010.5132], providing the complete proof of definitive boundedness and decay results for the scalar wave equation on Kerr backgroun…

New approach simplifies proof of wave equations on black holes.

EPGP surrogate outperforms finite elements in solving wave equations.

We recover the higher order terms for the acoustic wave equation from measurements of the modulus of the solution. The recovery of these coefficients is reduced to a question of stability for inverting a Hamiltonian flow transform, not the geodesic X-ray transform encountered in other inverse boundary problems like the…

We obtain necessary and sufficient conditions for the existence of "conservation laws" on null hypersurfaces for the wave equation on general four-dimensional Lorentzian manifolds. Examples of null hypersurfaces exhibiting such conservation laws include the standard null cones of Minkowski spacetime and the degenerate …