scholarly journals Time exponential splitting integrator for the Klein–Gordon equation with free parameters in the Hagstrom–Warburton absorbing boundary conditions

2018 ◽  
Vol 333 ◽  
pp. 185-199 ◽  
Author(s):  
I. Alonso-Mallo ◽  
A.M. Portillo
Keyword(s):  
Boundary Conditions ◽  
Gordon Equation ◽  
Absorbing Boundary Conditions ◽  
Absorbing Boundary ◽  
Klein Gordon ◽  
Exponential Splitting ◽  
Free Parameters ◽  
Klein Gordon Equation

scholarly journals Theoretical Investigation of Subluminal Particles Endowed with Imaginary Mass

2021 ◽  
Author(s):  
Luca Nanni
Keyword(s):  
General Solution ◽  
Wave Packet ◽  
Boundary Conditions ◽  
Theoretical Investigation ◽  
Wave Equations ◽  
Gordon Equation ◽  
Fourier Integral ◽  
Klein Gordon ◽  
Group Velocities ◽  
Klein Gordon Equation

In this article, the general solution of the tachyonic Klein-Gordon equation is obtained as a Fourier integral performed on a suitable path in the complex \omega-plane. In particular, it is proved that under given boundary conditions this solution does not contain any superluminal components. On the basis of this result, we infer that all possible spacelike wave equations describe the dynamics of subluminal particles endowed with imaginary mass. This result is validated for the Chodos equation, used to describe the hypothetical superluminal behaviour of neutrino. In this specific framework, it is proved that the wave packet propagates in spacetime with subluminal group velocities and that for enough small energies it behaves as a localized wave.

scholarly journals Theoretical Investigation of Subluminal Particles Endowed with Imaginary Mass

Particles ◽  
2021 ◽  
Vol 4 (2) ◽  
pp. 325-332
Author(s):  
Luca Nanni
Keyword(s):  
General Solution ◽  
Wave Packet ◽  
Boundary Conditions ◽  
Wave Equations ◽  
Gordon Equation ◽  
Fourier Integral ◽  
Klein Gordon ◽  
The Given ◽  
Group Velocities ◽  
Klein Gordon Equation

In this article, the general solution of the tachyonic Klein–Gordon equation is obtained as a Fourier integral performed on a suitable path in the complex ω-plane. In particular, it is proved that this solution does not contain any superluminal components under the given boundary conditions. On the basis of this result, we infer that all possible spacelike wave equations describe the dynamics of subluminal particles endowed with imaginary mass. This result is validated for the Chodos equation, used to describe the hypothetical superluminal behaviour of the neutrino. In this specific framework, it is proved that the wave packet propagates in spacetime with subluminal group velocities and that it behaves as a localized wave for sufficiently small energies.

scholarly journals The Massive Feynman Propagator on Asymptotically Minkowski Spacetimes II

2019 ◽  
Vol 2020 (20) ◽  
pp. 6856-6870
Author(s):  
Christian Gérard ◽  
Michał Wrochna
Keyword(s):  
Boundary Conditions ◽  
Recent Work ◽  
Short Range ◽  
Gordon Equation ◽  
Feynman Propagator ◽  
Klein Gordon ◽  
Type Boundary ◽  
Klein Gordon Equation

Abstract We consider the massive Klein–Gordon equation on short-range asymptotically Minkowski spacetimes. Extending our results in [7], we show that the Klein–Gordon operator with Feynman-type boundary conditions at infinite times is invertible and that its inverse, called the Feynman inverse, satisfies the microlocal conditions of Feynman parametrices in the sense of Duistermaat and Hörmander. This supplements the recent work of Vasy [10] with more explicit techniques.

Chaotic Oscillations of Solutions of the Klein–Gordon Equation Due to Imbalance of Distributed and Boundary Energy Flows

2014 ◽  
Vol 24 (07) ◽  
pp. 1430021 ◽  
Author(s):  
Goong Chen ◽  
Bo Sun ◽  
Tingwen Huang
Keyword(s):  
Boundary Conditions ◽  
Gordon Equation ◽  
Boundary Energy ◽  
Nonlinear Pdes ◽  
Van Der Pol ◽  
Energy Flows ◽  
Spatially Distributed ◽  
Klein Gordon ◽  
The One ◽  
Klein Gordon Equation

Chaos in physical systems governed by nonlinear PDEs have been amply observed and reported. However, rigorous proofs for their occurrences are challenging. In particular, for a second order PDE of hyperbolic type with a van der Pol cubic nonlinearity in one of the boundary conditions and a spatially distributed antidamping term in a linear governing equation, no proof for the onset of chaos was available even though chaos was expected to occur when the antidamping term becomes sufficiently strong. In this paper, we use an operator-factoring technique together with the analogy with the one-dimensional wave equation to prove that for the Klein–Gordon equation chaos occurs for a class of equations and boundary conditions when system parameters enter a certain regime. Chaotic and nonchaotic profiles of solutions are illustrated by computer graphics.

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