arXiv research

New method generates optical vortices in any knot shape.

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

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

Classifies solutions to vacuum weighted Einstein equations on pr-waves.

Unique solutions found for wave-like decaying null infinity equations.

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

Geometric flow on curves in S^3 generates YO equations solutions.

High frequency limit for most of wave phenomena is known as quasiclassical limit or ray optics limit. Propagation of waves in this limit is described in terms of wave fronts and rays. Wave front is a surface of constant phase whose points are moving along rays. As it appears, their motion can be described by Hamilton e…

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

Study proves interaction of three impulsive gravitational waves, showing local solution and Lipschitz continuity.

The aim of this paper is to construct and analyze solutions to a class of Hamilton-Jacobi-Bellman equations with range bounds on the optimal response variable. Using the Riccati transformation we derive and analyze a fully nonlinear parabolic partial differential equation for the optimal response function. We construct…

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

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…

Measuring wave sources uniquely identifies manifold properties.

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.

Study of Randers spacetimes and their Finsler gravity solutions.

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…

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

Novel approach to wave equations near null infinity in flat spacetimes.

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 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.

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 $…

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 investigate the geometric propagation and diffraction of singularities of solutions to the wave equation on manifolds with edge singularities.

Solitary waves are localized gravity waves that preserve their consistency and henceforth their visibility through properties of nonlinear hydrodynamics. Solitary waves have finite amplitude and spread with constant speed and constant shape. In this paper, we have used Lie group of transformation method to solve (3 + 1…

Geometric focusing affects dispersive estimates for Schrödinger and wave equations.

Spectral embedding uses eigenfunctions of the discrete Laplacian on a weighted graph to obtain coordinates for an embedding of an abstract data set into Euclidean space. We propose a new pre-processing step of first using the eigenfunctions to simulate a low-frequency wave moving over the data and using both position a…

Researchers find counterexamples to inverse problems for wave equations.

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

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.

Any surface can be foliated into equipotential hypersurfaces of the level sets. A current result is that the contours are the progressing wave fronts of a certain hyperbolic partial differential equation, a wave equation. It is connected with the gradient lines, as well as with a corresponding eikonal equation. The lev…

A nonlinear wave alternative for the standard Black-Scholes option-pricing model is presented. The adaptive-wave model, representing 'controlled Brownian behavior' of financial markets, is formally defined by adaptive nonlinear Schrödinger (NLS) equations, defining the option-pricing wave function in terms of the stock…

Consider the equivariant wave map equation from Minkowski space to a rotationnally symmetric manifold which has an equator (example: the sphere). In dimension 3, this article gives a necessary and sufficient condition for the existence of a smooth self-similar blow up profile. More generally, we study the relation betw…

We give new results concerning the Frobenius integrability and solution of evolution equations admitting travelling wave solutions. In particular, we give a powerful result which explains the extraordinary integrability of some of these equations. We also discuss "local" conservations laws for evolution equations in ge…

A global weak solution of the biharmonic wave map equation in the energy space for spherical targets is constructed. The equation is reformulated as a conservation law and solved by a suitable Ginzburg-Landau type approximation.

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

We study the wave analog of the Liouville equation and the constant mean curvature equations in 2 space dimensions, which are energy critical. We exhibit a blow-up criteria for the former using tools from conformal geometry, and we exhibit finite time blow-up for the latter under suitable assumptions on the initial dat…

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

Second paper in series solves Einstein vacuum equations for three impulsive waves.

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…

Wave maps from circle to manifold controllable if homotopy classes match.

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

Analyzes properties of stiffness tensors for elastic wave imaging.

We review some recent results on geometric equations on Lorentzian manifolds such as the wave and Dirac equations. This includes well-posedness and stability for various initial value problems, as well as results on the structure of these equations on black-hole spacetimes (in particular, on the Kerr solution), the ind…

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…

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.

Study on wave-breaking phenomena in solutions of the Camassa-Holm equation.