arXiv research

Study peels tensor equations on Schwarzschild spacetime.

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

Study proves global existence and decay for complex wave equations.

We derive a representation formula for the tensorial wave equation $\Box_\bg φ^I=F^I$ in globally hyperbolic Lorentzian spacetimes $(\M^{2+1}, \bg)$ by giving a geometric formulation of the method of descent which is applicable for any dimension.

Proves stability of Minkowski space-time for Einstein-Yang-Mills equations.

Tensor networks and RNNs are equivalent, improving wave function encoding.

New metrics derived from geodesics simplify semi-Riemannian geometry.

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

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

Developed a new symmetric hyperbolic formulation for Einstein-Yang-Mills system.

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.

Traditional Recurrent Neural Networks assume vectorized data as inputs. However many data from modern science and technology come in certain structures such as tensorial time series data. To apply the recurrent neural networks for this type of data, a vectorisation process is necessary, while such a vectorisation leads…

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…

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

Paper analyzes online tensorial ICA convergence with stochastic approximation.

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

A new tensorial metric describes geometry in 4D space.

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, 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 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.

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…

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…

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

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.

We give a tensorial description of the Turaev cobracket on any genus 0 compact surface through the standard group-like expansion, where the Bernoulli numbers appear.

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…

The paper defines constraints for commuting endomorphisms in generalized tangent bundles.

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…