scholarly journals Wave equation on spherically symmetric Lorentzian metrics

2011 ◽  
Vol 52 (6) ◽  
pp. 063511 ◽  
Author(s):  
Ashfaque H. Bokhari ◽  
Ahmad Y. Al-Dweik ◽  
A. H. Kara ◽  
M. Karim ◽  
F. D. Zaman
Keyword(s):  
Wave Equation ◽  
Spherically Symmetric ◽  
Lorentzian Metrics

Isospectral sets for transmission eigenvalue problem

2020 ◽  
Vol 28 (1) ◽  
pp. 63-69 ◽  
Author(s):  
Chuan-Fu Yang ◽  
Sergey A. Buterin
Keyword(s):  
Wave Equation ◽  
Eigenvalue Problem ◽  
Wave Speed ◽  
Spherically Symmetric ◽  
Variable Speed ◽  
Transmission Eigenvalues ◽  
Constructive Methods ◽  
Transmission Eigenvalue ◽  
Symmetric Variable ◽  
Transmission Eigenvalue Problem

AbstractWe consider the boundary value problem {R(a,q)}: {-y^{\prime\prime}(x)+q(x)y(x)=\lambda y(x)} with {y(0)=0} and {y(1)\cos(a\sqrt{\lambda})=y^{\prime}(1)\frac{\sin(a\sqrt{\lambda})}{\sqrt{% \lambda}}}. Motivated by the previous work [T. Aktosun and V. G. Papanicolaou, Reconstruction of the wave speed from transmission eigenvalues for the spherically symmetric variable-speed wave equation, Inverse Problems 29 2013, 6, Article ID 065007], it is natural to consider the following interesting question: how does one characterize isospectral sets corresponding to problem {R(1,q)}? In this paper applying constructive methods we answer the above question.

Numerical Determination of Pseudobreathers of a Three-Dimensional Spherically Symmetric Wave Equation

2016 ◽  
Vol 12 (3) ◽  
Author(s):  
Janusz Karkowski
Keyword(s):  
Wave Equation ◽  
Three Dimensional ◽  
Spherically Symmetric ◽  
Numerical Determination ◽  
Pseudospectral Methods ◽  
Symmetric Wave ◽  
Long Time ◽  
Signum Function ◽  
Continuous Approximations

A numerical method for finding spherically symmetric pseudobreathers of a nonlinear wave equation is presented. The algorithm, based on pseudospectral methods, is applied to find quasi-periodic solutions with force terms being continuous approximations of the signum function. The obtained pseudobreathers slowly radiate energy and decay after some (usually long) time depending on the period that characterizes (unambiguously) the initial configuration.

scholarly journals GLOBAL REGULARITY FOR A LOGARITHMICALLY SUPERCRITICAL DEFOCUSING NONLINEAR WAVE EQUATION FOR SPHERICALLY SYMMETRIC DATA

2007 ◽  
Vol 04 (02) ◽  
pp. 259-265 ◽  
Author(s):  
TERENCE TAO
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Global Regularity ◽  
Spherically Symmetric ◽  
Supercritical Regime ◽  
Critical Equation ◽  
Critical Regularity ◽  
Spatial Dimensions ◽  
Symmetric Initial Data ◽  
Supercritical Wave Equation

We establish global regularity for the logarithmically energy-supercritical wave equation □u = u5 log (2 + u2) in three spatial dimensions for spherically symmetric initial data, by modifying an argument of Ginibre, Soffer and Velo for the energy-critical equation. This example demonstrates that critical regularity arguments can penetrate very slightly into the supercritical regime.

scholarly journals The Case Against Smooth Null Infinity I: Heuristics and Counter-Examples

2021 ◽  
Author(s):  
Leonhard M. A. Kehrberger
Keyword(s):  
Asymptotic Expansion ◽  
Wave Equation ◽  
Gravitational Radiation ◽  
Scattering Problem ◽  
Scattering Data ◽  
Linear Wave ◽  
Spherically Symmetric ◽  
Field Equations ◽  
Null Infinity ◽  
World Scientific Publishing

AbstractThis paper initiates a series of works dedicated to the rigorous study of the precise structure of gravitational radiation near infinity. We begin with a brief review of an argument due to Christodoulou (in: The Ninth Marcel Grossmann Meeting, World Scientific Publishing Company, Singapore, 2002) stating that Penrose’s proposal of smooth conformal compactification of spacetime (or smooth null infinity) fails to accurately capture the structure of gravitational radiation emitted by N infalling masses coming from past timelike infinity $$i^-$$ i - . Modelling gravitational radiation by scalar radiation, we then take a first step towards a dynamical understanding of the non-smoothness of null infinity by constructing solutions to the spherically symmetric Einstein–Scalar field equations that arise from polynomially decaying boundary data, $$r\phi \sim t^{-1}$$ r ϕ ∼ t - 1 as $$t\rightarrow -\infty $$ t → - ∞ , on a timelike hypersurface (to be thought of as the surface of a star) and the no incoming radiation condition, $$r\partial _v\phi =0$$ r ∂ v ϕ = 0 , on past null infinity. We show that if the initial Hawking mass at $$i^-$$ i - is nonzero, then, in accordance with the non-smoothness of $${\mathcal {I}}^+$$ I + , the asymptotic expansion of $$\partial _v(r\phi )$$ ∂ v ( r ϕ ) near $${\mathcal {I}}^+$$ I + reads $$\partial _v(r\phi )=Cr^{-3}\log r+{\mathcal {O}}(r^{-3})$$ ∂ v ( r ϕ ) = C r - 3 log r + O ( r - 3 ) for some non-vanishing constant C. In fact, the same logarithmic terms appear already in the linear theory, i.e. when considering the spherically symmetric linear wave equation on a fixed Schwarzschild background. As a corollary, we can apply our results to the scattering problem on Schwarzschild: Putting compactly supported scattering data for the linear (or coupled) wave equation on $${\mathcal {I}}^-$$ I - and on $${\mathcal {H}}^-$$ H - , we find that the asymptotic expansion of $$\partial _v(r\phi )$$ ∂ v ( r ϕ ) near $${\mathcal {I}}^+$$ I + generically contains logarithmic terms at second order, i.e. at order $$r^{-4}\log r$$ r - 4 log r .

Asymptotic theory of body waves in a radially heterogeneous Earth

1969 ◽  
Vol 59 (5) ◽  
pp. 2039-2059
Author(s):  
Sarva Jit Singh ◽  
Ari Ben-Menahem
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
High Frequency ◽  
Asymptotic Theory ◽  
Heterogeneous Medium ◽  
Body Waves ◽  
Spherically Symmetric ◽  
Vector Wave ◽  
Vector Wave Equation ◽  
Earth Models

abstract Various aspects of elastic wave propagation in a spherically symmetric, non-gravitating, isotropic, inhomogeneous medium are considered. It is shown through a simple example that the high frequency decoupling conditions of the vector wave equation may be approximately satisfied by real Earth models. An asymptotic theory is developed for the decoupled potential equations. This theory is applied to the case of a shear dislocation and to that of a center of compression in a radially heterogeneous medium. Explicit expressions are obtained for the ray-theoretical displacements.

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