11,485 research outputs found
A stability analysis of a real space split operator method for the Klein-Gordon equation
We carry out a stability analysis for the real space split operator method
for the propagation of the time-dependent Klein-Gordon equation that has been
proposed Ruf et al. [M. Ruf, H. Bauke, C.H. Keitel, A real space split operator
method for the Klein-Gordon equation, Journal of Computational Physics 228 (24)
(2009) 9092-9106, doi:10.1016/j.jcp.2009.09.012]. The region of algebraic
stability is determined analytically by means of a von-Neumann stability
analysis for systems with homogeneous scalar and vector potentials. Algebraic
stability implies convergence ofthe real space split operator method for smooth
absolutely integrable initial conditions. In the limit of small spatial grid
spacings in each of the spatial dimensions and small temporal steps
, the stability condition becomes for second order
finite differences and for fourth order finite
differences, respectively, with denoting the speed of light. Furthermore,
we demonstrate numerically that the stability region for systems with
inhomogeneous potentials coincides almost with the region of algebraic
stability for homogeneous potentials
Computational relativistic quantum dynamics and its application to relativistic tunneling and Kapitza-Dirac scattering
Computational methods are indispensable to study the quantum dynamics of
relativistic light-matter interactions in parameter regimes where analytical
methods become inapplicable. We present numerical methods for solving the
time-dependent Dirac equation and the time-dependent Klein-Gordon equation and
their implementation on high performance graphics cards. These methods allow us
to study tunneling from hydrogen-like highly charged ions in strong laser
fields and Kapitza-Dirac scattering in the relativistic regime
Normal form for travelling kinks in discrete Klein-Gordon lattices
We study travelling kinks in the spatial discretizations of the nonlinear
Klein--Gordon equation, which include the discrete lattice and the
discrete sine--Gordon lattice. The differential advance-delay equation for
travelling kinks is reduced to the normal form, a scalar fourth-order
differential equation, near the quadruple zero eigenvalue. We show numerically
non-existence of monotonic kinks (heteroclinic orbits between adjacent
equilibrium points) in the fourth-order equation. Making generic assumptions on
the reduced fourth-order equation, we prove the persistence of bounded
solutions (heteroclinic connections between periodic solutions near adjacent
equilibrium points) in the full differential advanced-delay equation with the
technique of center manifold reduction. Existence and persistence of multiple
kinks in the discrete sine--Gordon equation are discussed in connection to
recent numerical results of \cite{ACR03} and results of our normal form
analysis
Wave and Klein-Gordon equations on hyperbolic spaces
We consider the Klein--Gordon equation associated with the Laplace--Beltrami
operator on real hyperbolic spaces of dimension ; as
has a spectral gap, the wave equation is a particular case of our
study. After a careful kernel analysis, we obtain dispersive and Strichartz
estimates for a large family of admissible couples. As an application, we prove
global well--posedness results for the corresponding semilinear equation with
low regularity data.Comment: 50 pages, 30 figures. arXiv admin note: text overlap with
arXiv:1010.237
Multi-site breathers in Klein-Gordon lattices: stability, resonances, and bifurcations
We prove the most general theorem about spectral stability of multi-site
breathers in the discrete Klein-Gordon equation with a small coupling constant.
In the anti-continuum limit, multi-site breathers represent excited
oscillations at different sites of the lattice separated by a number of "holes"
(sites at rest). The theorem describes how the stability or instability of a
multi-site breather depends on the phase difference and distance between the
excited oscillators. Previously, only multi-site breathers with adjacent
excited sites were considered within the first-order perturbation theory. We
show that the stability of multi-site breathers with one-site holes change for
large-amplitude oscillations in soft nonlinear potentials. We also discover and
study a symmetry-breaking (pitchfork) bifurcation of one-site and multi-site
breathers in soft quartic potentials near the points of 1:3 resonance.Comment: 34 pages, 12 figure
Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm
We show that the Maxwell-Klein-Gordon equations in three dimensions are
globally well-posed in in the Coulomb gauge for all . This extends previous work of Klainerman-Machedon
\cite{kl-mac:mkg} on finite energy data , and Eardley-Moncrief
\cite{eardley} for still smoother data. We use the method of almost
conservation laws, sometimes called the "I-method", to construct an almost
conserved quantity based on the Hamiltonian, but at the regularity of
rather than . One then uses Strichartz, null form, and commutator
estimates to control the development of this quantity. The main technical
difficulty (compared with other applications of the method of almost
conservation laws) is at low frequencies, because of the poor control on the
norm. In an appendix, we demonstrate the equations' relative lack of
smoothing - a property that presents serious difficulties for studying rough
solutions using other known methods.Comment: 56 pages, no figures; to appear, DCDS-A. This is the final version,
incorporating the referee comment
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