8,143 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
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
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
On scattering of solitons for the Klein-Gordon equation coupled to a particle
We establish the long time soliton asymptotics for the translation invariant
nonlinear system consisting of the Klein-Gordon equation coupled to a charged
relativistic particle. The coupled system has a six dimensional invariant
manifold of the soliton solutions. We show that in the large time approximation
any finite energy solution, with the initial state close to the solitary
manifold, is a sum of a soliton and a dispersive wave which is a solution of
the free Klein-Gordon equation. It is assumed that the charge density satisfies
the Wiener condition which is a version of the ``Fermi Golden Rule''. The proof
is based on an extension of the general strategy introduced by Soffer and
Weinstein, Buslaev and Perelman, and others: symplectic projection in Hilbert
space onto the solitary manifold, modulation equations for the parameters of
the projection, and decay of the transversal component.Comment: 47 pages, 2 figure
Stable directions for small nonlinear Dirac standing waves
We prove that for a Dirac operator with no resonance at thresholds nor
eigenvalue at thresholds the propagator satisfies propagation and dispersive
estimates. When this linear operator has only two simple eigenvalues close
enough, we study an associated class of nonlinear Dirac equations which have
stationary solutions. As an application of our decay estimates, we show that
these solutions have stable directions which are tangent to the subspaces
associated with the continuous spectrum of the Dirac operator. This result is
the analogue, in the Dirac case, of a theorem by Tsai and Yau about the
Schr\"{o}dinger equation. To our knowledge, the present work is the first
mathematical study of the stability problem for a nonlinear Dirac equation.Comment: 62 page
PT-symmetric sine-Gordon breathers
In this work, we explore a prototypical example of a genuine continuum
breather (i.e., not a standing wave) and the conditions under which it can
persist in a -symmetric medium. As our model of interest, we
will explore the sine-Gordon equation in the presence of a -
symmetric perturbation. Our main finding is that the breather of the
sine-Gordon model will only persist at the interface between gain and loss that
-symmetry imposes but will not be preserved if centered at the
lossy or at the gain side. The latter dynamics is found to be interesting in
its own right giving rise to kink-antikink pairs on the gain side and complete
decay of the breather on the lossy side. Lastly, the stability of the breathers
centered at the interface is studied. As may be anticipated on the basis of
their "delicate" existence properties such breathers are found to be
destabilized through a Hopf bifurcation in the corresponding Floquet analysis
Orbital stability of periodic waves in the class of reduced Ostrovsky equations
Periodic travelling waves are considered in the class of reduced Ostrovsky
equations that describe low-frequency internal waves in the presence of
rotation. The reduced Ostrovsky equations with either quadratic or cubic
nonlinearities can be transformed to integrable equations of the Klein--Gordon
type by means of a change of coordinates. By using the conserved momentum and
energy as well as an additional conserved quantity due to integrability, we
prove that small-amplitude periodic waves are orbitally stable with respect to
subharmonic perturbations, with period equal to an integer multiple of the
period of the wave. The proof is based on construction of a Lyapunov
functional, which is convex at the periodic wave and is conserved in the time
evolution. We also show numerically that convexity of the Lyapunov functional
holds for periodic waves of arbitrary amplitudes.Comment: 34 page
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