10,645 research outputs found
Excitation Thresholds for Nonlinear Localized Modes on Lattices
Breathers are spatially localized and time periodic solutions of extended
Hamiltonian dynamical systems. In this paper we study excitation thresholds for
(nonlinearly dynamically stable) ground state breather or standing wave
solutions for networks of coupled nonlinear oscillators and wave equations of
nonlinear Schr\"odinger (NLS) type. Excitation thresholds are rigorously
characterized by variational methods. The excitation threshold is related to
the optimal (best) constant in a class of discr ete interpolation inequalities
related to the Hamiltonian energy. We establish a precise connection among ,
the dimensionality of the lattice, , the degree of the nonlinearity
and the existence of an excitation threshold for discrete nonlinear
Schr\"odinger systems (DNLS).
We prove that if , then ground state standing waves exist if
and only if the total power is larger than some strictly positive threshold,
. This proves a conjecture of Flach, Kaldko& MacKay in
the context of DNLS. We also discuss upper and lower bounds for excitation
thresholds for ground states of coupled systems of NLS equations, which arise
in the modeling of pulse propagation in coupled arrays of optical fibers.Comment: To appear in Nonlinearit
Template-based Gravitational-Wave Echoes Search Using Bayesian Model Selection
The ringdown of the gravitational-wave signal from a merger of two black
holes has been suggested as a probe of the structure of the remnant compact
object, which may be more exotic than a black hole. It has been pointed out
that there will be a train of echoes in the late-time ringdown stage for
different types of exotic compact objects. In this paper, we present a
template-based search methodology using Bayesian statistics to search for
echoes of gravitational waves. Evidence for the presence or absence of echoes
in gravitational-wave events can be established by performing Bayesian model
selection. The Occam factor in Bayesian model selection will automatically
penalize the more complicated model that echoes are present in
gravitational-wave strain data because of its higher degree of freedom to fit
the data. We find that the search methodology was able to identify
gravitational-wave echoes with Abedi et al.'s echoes waveform model about 82.3%
of the time in simulated Gaussian noise in the Advanced LIGO and Virgo network
and about 61.1% of the time in real noise in the first observing run of
Advanced LIGO with significance. Analyses using this method are
performed on the data of Advanced LIGO's first observing run, and we find no
statistical significant evidence for the detection of gravitational-wave
echoes. In particular, we find combined evidence of the three events
in Advanced LIGO's first observing run. The analysis technique developed in
this paper is independent of the waveform model used, and can be used with
different parametrized echoes waveform models to provide more realistic
evidence of the existence of echoes from exotic compact objects.Comment: 16 pages, 6 figure
Geometry of Non-Hausdorff Spaces and Its Significance for Physics
Hausdorff relation, topologically identifying points in a given space,
belongs to elementary tools of modern mathematics. We show that if subtle
enough mathematical methods are used to analyze this relation, the conclusions
may be far-reaching and illuminating. Examples of situations in which the
Hausdorff relation is of the total type, i.e., when it identifies all points of
the considered space, are the space of Penrose tilings and space-times of some
cosmological models with strong curvature singularities. With every Hausdorff
relation a groupoid can be associated, and a convolutive algebra defined on it
allows one to analyze the space that otherwise would remain intractable. The
regular representation of this algebra in a bundle of Hilbert spaces leads to a
von Neumann algebra of random operators. In this way, a probabilistic
description (in a generalized sense) naturally takes over when the concept of
point looses its meaning. In this situation counterparts of the position and
momentum operators can be defined, and they satisfy a commutation relation
which, in the suitable limiting case, reproduces the Heisenberg indeterminacy
relation. It should be emphasized that this is neither an additional assumption
nor an effect of a quantization process, but simply the consequence of a purely
geometric analysis.Comment: 13 LaTex pages, no figure
On Deusons or Deuteronlike Meson-Meson Bound States
The systematics of deuteronlike two-meson bound states, {\it deusons}, is
discussed. Previous arguments that many of the present non- states are
such states are elaborated including, in particular, the tensor potential. For
pseudoscalar states the important observation is made that the centrifugal
barrier from the P-wave can be overcome by the and terms of the
tensor potential. In the heavy meson sector one-pion exchange alone is strong
enough to form at least deuteron-like and composites
bound by approximately 50 MeV, while and states are
expected near the threshold.Comment: Invited talk at the Hadron93 International Conf. on Hadron
Spectroscopy, Como, Italy 22.-25.6. 1993. 5 pages in LATEX HU-SEFT R 1993-13
Generalized Entanglement as a Natural Framework for Exploring Quantum Chaos
We demonstrate that generalized entanglement [Barnum {\em et al.}, Phys. Rev.
A {\bf 68}, 032308 (2003)] provides a natural and reliable indicator of quantum
chaotic behavior. Since generalized entanglement depends directly on a choice
of preferred observables, exploring how generalized entanglement increases
under dynamical evolution is possible without invoking an auxiliary coupled
system or decomposing the system into arbitrary subsystems. We find that, in
the chaotic regime, the long-time saturation value of generalized entanglement
agrees with random matrix theory predictions. For our system, we provide
physical intuition into generalized entanglement within a single system by
invoking the notion of extent of a state. The latter, in turn, is related to
other signatures of quantum chaos.Comment: clarified and expanded version accepted by Europhys. Let
Stability and symmetry-breaking bifurcation for the ground states of a NLS with a interaction
We determine and study the ground states of a focusing Schr\"odinger equation
in dimension one with a power nonlinearity and a strong
inhomogeneity represented by a singular point perturbation, the so-called
(attractive) interaction, located at the origin. The
time-dependent problem turns out to be globally well posed in the subcritical
regime, and locally well posed in the supercritical and critical regime in the
appropriate energy space. The set of the (nonlinear) ground states is
completely determined. For any value of the nonlinearity power, it exhibits a
symmetry breaking bifurcation structure as a function of the frequency (i.e.,
the nonlinear eigenvalue) . More precisely, there exists a critical
value \om^* of the nonlinear eigenvalue \om, such that: if \om_0 < \om <
\om^*, then there is a single ground state and it is an odd function; if \om
> \om^* then there exist two non-symmetric ground states. We prove that before
bifurcation (i.e., for \om < \om^*) and for any subcritical power, every
ground state is orbitally stable. After bifurcation (\om =\om^*+0), ground
states are stable if does not exceed a value that lies
between 2 and 2.5, and become unstable for . Finally, for and \om \gg \om^*, all ground states are unstable. The branch of odd
ground states for \om \om^*,
obtaining a family of orbitally unstable stationary states. Existence of ground
states is proved by variational techniques, and the stability properties of
stationary states are investigated by means of the Grillakis-Shatah-Strauss
framework, where some non standard techniques have to be used to establish the
needed properties of linearization operators.Comment: 46 pages, 5 figure
Four-quark state in QCD
The spectra of some 0++ four-quark states, which are composed of \bar qq
pairs, are calculated in QCD. The light four-quark states are calculated using
the traditional sum rules while four-quark states containing one heavy quark
are computed in HQET. For constructing the interpolating currents, different
couplings of the color and spin inside the \bar qq pair are taken into account.
It is found that the spin and color combination has little effect on the mass
of the four-quark states.Comment: 10 pages, 4 ps figures, Late
Degenerate dispersive equations arising in the study of magma dynamics
An outstanding problem in Earth science is understanding the method of
transport of magma in the Earth's mantle. Models for this process, transport in
a viscously deformable porous media, give rise to scalar degenerate,
dispersive, nonlinear wave equations. We establish a general local
well-posedness for a physical class of data (roughly ) via fixed point
methods. The strategy requires positive lower bounds on the solution. This is
extended to global existence for a subset of possible nonlinearities by making
use of certain conservation laws associated with the equations. Furthermore, we
construct a Lyapunov energy functional, which is locally convex about the
uniform state, and prove (global in time) nonlinear dynamic stability of the
uniform state for any choice of nonlinearity. We compare the dynamics to that
of other problems and discuss open questions concerning a larger range of
nonlinearities, for which we conjecture global existence.Comment: 27 Pages, 7 figures are not present in this version. See
http://www.columbia.edu/~grs2103/ for a PDF with figures. Submitted to
Nonlinearit
On the topology of stationary black holes
We prove that the domain of outer communication of a stationary, globally
hyperbolic spacetime satisfying the null energy condition must be simply
connected. Under suitable additional hypotheses, this implies, in particular,
that each connected component of a cross-section of the event horizon of a
stationary black hole must have spherical topology.Comment: 7 pages, Late
Zero curvature representation for classical lattice sine-Gordon equation via quantum R-matrix
Local M-operators for the classical sine-Gordon model in discrete space-time
are constructed by convolution of the quantum trigonometric 44 R-matrix
with certain vectors in its "quantum" space. Components of the vectors are
identified with -functions of the model. This construction generalizes
the known representation of M-operators in continuous time models in terms of
Lax operators and classical -matrix.Comment: 10 pages, LaTeX (misprints are corrected
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