4,270 research outputs found
R-adaptive multisymplectic and variational integrators
Moving mesh methods (also called r-adaptive methods) are space-adaptive
strategies used for the numerical simulation of time-dependent partial
differential equations. These methods keep the total number of mesh points
fixed during the simulation, but redistribute them over time to follow the
areas where a higher mesh point density is required. There are a very limited
number of moving mesh methods designed for solving field-theoretic partial
differential equations, and the numerical analysis of the resulting schemes is
challenging. In this paper we present two ways to construct r-adaptive
variational and multisymplectic integrators for (1+1)-dimensional Lagrangian
field theories. The first method uses a variational discretization of the
physical equations and the mesh equations are then coupled in a way typical of
the existing r-adaptive schemes. The second method treats the mesh points as
pseudo-particles and incorporates their dynamics directly into the variational
principle. A user-specified adaptation strategy is then enforced through
Lagrange multipliers as a constraint on the dynamics of both the physical field
and the mesh points. We discuss the advantages and limitations of our methods.
Numerical results for the Sine-Gordon equation are also presented.Comment: 65 pages, 13 figure
Non-Abelian Monopole in the Parameter Space of Point-like Interactions
We study non-Abelian geometric phase in supersymmetric
quantum mechanics for a free particle on a circle with two point-like
interactions at antipodal points. We show that non-Abelian Berry's connection
is that of magnetic monopole discovered by Moody, Shapere and Wilczek
in the context of adiabatic decoupling limit of diatomic molecule.Comment: 15 pages, 3 tikz figures; minor correction
Lifshitz quasinormal modes and relaxation from holography
We obtain relaxation times for field theories with Lifshitz scaling and with
holographic duals Einstein-Maxwell-Dilaton gravity theories. This is done by
computing quasinormal modes of a bulk scalar field in the presence of Lifshitz
black branes. We determine the relation between relaxation time and dynamical
exponent z, for various values of boundary dimension d and operator scaling
dimension. It is found that for d>z+1, at zero momenta, the modes are
non-overdamped, whereas for d<=z+1 the system is always overdamped. For d=z+1
and zero momenta, we present analytical results.Comment: 16 pages and 5 figure
Hamiltonian dynamics of a quantum of space: hidden symmetries and spectrum of the volume operator, and discrete orthogonal polynomials
The action of the quantum mechanical volume operator, introduced in
connection with a symmetric representation of the three-body problem and
recently recognized to play a fundamental role in discretized quantum gravity
models, can be given as a second order difference equation which, by a complex
phase change, we turn into a discrete Schr\"odinger-like equation. The
introduction of discrete potential-like functions reveals the surprising
crucial role here of hidden symmetries, first discovered by Regge for the
quantum mechanical 6j symbols; insight is provided into the underlying
geometric features. The spectrum and wavefunctions of the volume operator are
discussed from the viewpoint of the Hamiltonian evolution of an elementary
"quantum of space", and a transparent asymptotic picture emerges of the
semiclassical and classical regimes. The definition of coordinates adapted to
Regge symmetry is exploited for the construction of a novel set of discrete
orthogonal polynomials, characterizing the oscillatory components of
torsion-like modes.Comment: 13 pages, 5 figure
- …