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In this paper we study the stability of the unique continuation in the case of the wave equation with variable coefficients independent of time. We prove a logarithmic estimate in a arbitrary domain of ${\mathbb R}^{n+1}$, where all the…
In this paper we study the uniqueness and the increasing stability in the inverse source problem for electromagnetic waves in homogeneous and inhomogeneous media from boundary data at multiple wave numbers. For the unique determination of…
On compact Riemannian manifolds with chaotic geometries, specifically those exhibiting the random wave model conjectured by Berry, we derive heuristically a homogeneous kinetic wave equation that is universal for all such manifolds.
These notes are intended as an introduction to the question of unique continuation for the wave operator, and some of its applications. The general question is whether a solution to a wave equation in a domain, vanishing on a subdomain has…
We present an axiomatic/synthetic account of the Huygens Principle of wave fronts. The primitive notions are "touching", and (a weak notion of ) metric. The paper simplifies some of the exposition of the author's "Metric spaces and SDG",…
The wave equation (free boson) problem is studied from the viewpoint of the relations on the symplectic manifolds associated to the boundary induced by solutions. Unexpectedly there is still something to say on this simple, well-studied…
In this paper, we discuss how the concepts of Hamiltonian optics are internally connected to the scalar wave theory of light rays. It is shown that the solutions of the reduced wave equation are similar to Huygen's wavelets, and they can be…
In this article, we extensively develop Carleman estimates for the wave equation and give some applications. We focus on the case of an observation of the flux on a part of the boundary satisfying the Gamma conditions of Lions. We will then…
We consider wave equations on Lorentzian manifolds in case of low regularity. We first extend the classical solution theory to prove global unique solvability of the Cauchy problem for distributional data and right hand side on smooth…
A novel canonical Hamiltonian formalism is developed for long internal waves in a rotating environment. This includes the effects of background vorticity and shear on the waves. By restricting consideration to flows in hydrostatic balance,…
This expository note gives a digest version of Hormander's propagation of singularities theorem for the wave equation.
This paper analyzes inverse scattering for the one-dimensional Helmholtz equation in the case where the wave speed is piecewise constant. Scattering data recorded for an arbitrarily small interval of frequencies is shown to determine the…
A wave equation with mass term is studied for all particles and antiparticles of the first generation: electron and its neutrino, positron and antineutrino, quarks $u$ and $d$ with three states of color and antiquarks $\overline{u}$ and…
The paper studies the inferences of wave equations for electromagnetic fields when there are gravitational fields at the same time. In the description with the algebra of octonions, the inferences of wave equations are identical with that…
The main purpose of this paper is to review the progress that has taken place so far in the search for a single unifying principle that harmonizes (i) the wave and particle natures of matter and radiation, both at the quantum and the…
We generalize Lindeberg's proof of the central limit theorem to an invariance principle for arbitrary smooth functions of independent and weakly dependent random variables. The result is applied to get a similar theorem for smooth functions…
A general type of localized excitations, folded solitary waves and foldons, are defined and studied both analytically and graphically. The folded solitary waves and foldons may be "folded" in quite complicated ways and possess quite rich…
This is a study of the Euler equations for free surface water waves in the case of varying bathymetry, considering the problem in the shallow water scaling regime. In the case of rapidly varying periodic bottom boundaries this is a problem…
We present a simple and self-contained approach to establish the unique continuation property for some classical evolution equations of second order in a cylindrical domain. We namely discuss this property for wave, parabolic and…
The recent observation of gravitational waves, stimulates the question of the longtime evolution of the space-time fluctuations. Gravitational waves interact themselves through the nonlinear character of Einstein's equations of general…