Related papers: Quantitative Mode Stability for the Wave Equation …
We formulate a general method for the study of semiclassical-like dynamics in stable regions of a mixed phase-space, in order to theoretically study the dynamics of quantum accelerator modes. In the simplest case, this involves determining…
We investigate the stability and long-term behavior of spatially periodic plane waves in the complex Klein-Gordon equation under localized perturbations. Such perturbations render the wave neither localized nor periodic, placing its…
In this paper, we study the stability and instability of plane wave solutions to semilinear systems of wave equations satisfying the null condition. We identify a condition which allows us to prove the global nonlinear asymptotic stability…
We show that the energy of classical solutions to the wave equation with hyperbolic boundary condition (i.e., dynamic Wentzell boundary condition) and damping on the boundary decays like 1/t. In fact we allow mixed boundary conditions: a…
The quasilocal energy associated with a constant stationary time slice of the Kerr spacetime is presented. The calculations are based on a recent proposal \cite{by} in which quasilocal energy is derived from the Hamiltonian of spatially…
In this paper we introduce a finite-parameters feedback control algorithm for stabilizing solutions of various classes of damped nonlinear wave equations. Specifically, stabilization the zero steady state solution of initial boundary value…
A static rotating cosmic string metric is singular along a timelike line and fails to be globally hyperbolic; these features make it difficult to solve the wave equation by conventional energy methods. Working on a single angular mode at a…
We introduce a rigorous framework for defining quasinormal modes on stationary, asymptotically flat spacetimes as isolated eigenvalues of the infinitesimal generator of time translations. We consider time functions corresponding to a…
We consider the nonlinear Schr{\"o}dinger equation with a harmonic potential in the presence of two combined energy-subcritical power nonlinearities. We assume that the larger power is defocusing, and the smaller power is focusing. Such a…
In this paper, we consider the subcritical half-wave equation in one dimension. Let $R_k(t,x)$, $k=1,2$, represent two-solitary wave solutions of the half-wave equation, each with different translations $x_1,x_2$. We prove that if the…
We discuss the (in)stability of solitary waves for a quasi-linear Schr{\"o}dinger equation. The equation contains a quasi-linear term, responsible for a saturation effect, as well as a power nonlinearity. For different exponents of the…
The purpose of this paper is to investigate the stabilization of a one-dimensional coupled wave equations with non smooth localized viscoelastic damping of Kelvin-Voigt type and localized time delay. Using a general criteria of…
It has been known classically that a star with an ergoregion but no event horizon is unstable to the emission of scalar, electromagnetic and gravitational waves. This classical ergoregion instability is characterized by complex frequency…
We present results from a new code for computing gravitational perturbations of the Kerr geometry. This new code carefully maintains high precision to allow us to obtain high-accuracy solutions for the gravitational quasinormal modes of the…
The wave equation $\square_{g_{M,a}}\psi=0$ on subextremal Kerr spacetimes $(\mathcal{M}_{M,a},g_{M,a})$, $0<|a|<M$, does not admit real mode solutions, as was established by Shlapentokh-Rothman. In this paper, we show that the absence of…
In the present work, we introduce a new $\mathcal{PT}$-symmetric variant of the Klein-Gordon field theoretic problem. We identify the standing wave solutions of the proposed class of equations and analyze their stability. In particular, we…
In this paper, we study the numerical stabilization of a 1D system of two wave equations coupled by velocities with an internal, local control acting on only one equation. In the theoretical part of this study, we distinguished two cases.…
We consider a stabilized finite element method based on a spacetime formulation, where the equations are solved on a global (unstructured) spacetime mesh. A unique continuation problem for the wave equation is considered, where data is…
We propose a model using the Korteweg-de Vries $(KdV)$ equation on a finite star-shaped network. We first prove the well-posedness of the system and give some regularity results. Then we prove that the energy of the solutions of the…
The Cauchy problem for the scalar wave equation in the Kerr geometry is considered, with initial data which is smooth and compactly supported outside the event horizon. A time-independent energy estimate for the outgoing wave is obtained.…