21,025 research outputs found
Infinity in string cosmology: A review through open problems
We review recent developments in the field of string cosmology with
particular emphasis on open problems having to do mainly with geometric
asymptotics and singularities. We discuss outstanding issues in a variety of
currently popular themes, such as tree-level string cosmology asymptotics,
higher-order string correction effects, M-theory cosmology, braneworlds, and
finally ambient cosmology.Comment: 37 pages, to appear in the IJMPD, v2: matches published versio
Ultraviolet asymptotics for quasiperiodic AdS_4 perturbations
Spherically symmetric perturbations in AdS-scalar field systems of small
amplitude epsilon approximately periodic on time scales of order 1/epsilon^2
(in the sense that no significant transfer of energy between the AdS normal
modes occurs) have played an important role in considerations of AdS stability.
They are seen as anchors of stability islands where collapse of small
perturbations to black holes does not occur. (This collapse, if it happens,
typically develops on time scales of the order 1/epsilon^2.) We construct an
analytic treatment of the frequency spectra of such quasiperiodic
perturbations, paying special attention to the large frequency asymptotics. For
the case of a self-interacting phi^4 scalar field in a non-dynamical AdS
background, we arrive at a fairly complete analytic picture involving
quasiperiodic spectra with an exponential suppression modulated by a power law
at large mode numbers. For the case of dynamical gravity, the structure of the
large frequency asymptotics is more complicated. We give analytic explanations
for the general qualitative features of quasiperiodic solutions localized
around a single mode, in close parallel to our discussion of the probe scalar
field, and find numerical evidence for logarithmic modulations in the
gravitational quasiperiodic spectra existing on top of the formulas previously
reported in the literature.Comment: 18 pages; v3: minor improvements, published versio
Detailed ultraviolet asymptotics for AdS scalar field perturbations
We present a range of methods suitable for accurate evaluation of the leading
asymptotics for integrals of products of Jacobi polynomials in limits when the
degrees of some or all polynomials inside the integral become large. The
structures in question have recently emerged in the context of effective
descriptions of small amplitude perturbations in anti-de Sitter (AdS)
spacetime. The limit of high degree polynomials corresponds in this situation
to effective interactions involving extreme short-wavelength modes, whose
dynamics is crucial for the turbulent instabilities that determine the ultimate
fate of small AdS perturbations. We explicitly apply the relevant asymptotic
techniques to the case of a self-interacting probe scalar field in AdS and
extract a detailed form of the leading large degree behavior, including closed
form analytic expressions for the numerical coefficients appearing in the
asymptotics.Comment: v2: 19 pages, expanded version accepted to JHE
Symmetry-breaking instability in a prototypical driven granular gas
Symmetry-breaking instability of a laterally uniform granular cluster (strip
state) in a prototypical driven granular gas is investigated. The system
consists of smooth hard disks in a two-dimensional box, colliding inelastically
with each other and driven, at zero gravity, by a "thermal" wall. The limit of
nearly elastic particle collisions is considered, and granular hydrodynamics
with the Jenkins-Richman constitutive relations is employed. The hydrodynamic
problem is completely described by two scaled parameters and the aspect ratio
of the box. Marginal stability analysis predicts a spontaneous symmetry
breaking instability of the strip state, similar to that predicted recently for
a different set of constitutive relations. If the system is big enough, the
marginal stability curve becomes independent of the details of the boundary
condition at the driving wall. In this regime, the density perturbation is
exponentially localized at the elastic wall opposite to the thermal wall. The
short- and long-wavelength asymptotics of the marginal stability curves are
obtained analytically in the dilute limit. The physics of the symmetry-breaking
instability is discussed.Comment: 11 pages, 14 figure
Forced Burgers Equation in an Unbounded Domain
The inviscid Burgers equation with random and spatially smooth forcing is
considered in the limit when the size of the system tends to infinity. For the
one-dimensional problem, it is shown both theoretically and numerically that
many of the features of the space-periodic case carry over to infinite domains
as intermediate time asymptotics. In particular, for large time we
introduce the concept of -global shocks replacing the notion of main shock
which was considered earlier in the periodic case (1997, E et al., Phys. Rev.
Lett. 78, 1904). In the case of spatially extended systems these objects are no
anymore global. They can be defined only for a given time scale and their
spatial density behaves as for large . The
probability density function of the age of shocks behaves
asymptotically as . We also suggest a simple statistical model for
the dynamics and interaction of shocks and discuss an analogy with the problem
of distribution of instability islands for a simple first-order stochastic
differential equation.Comment: 9 pages, 10 figures, revtex4, J. Stat. Phys, in pres
Stable topological textures in a classical 2D Heisenberg model
We show that stable localized topological soliton textures (skyrmions) with
topological charge exist in a classical 2D Heisenberg
model of a ferromagnet with uniaxial anisotropy. For this model the soliton
exist only if the number of bound magnons exceeds some threshold value depending on and the effective anisotropy constant .
We define soliton phase diagram as the dependence of threshold energies and
bound magnons number on anisotropy constant. The phase boundary lines are
monotonous for both and , while the solitons with
reveal peculiar nonmonotonous behavior, determining the transition regime from
low to high topological charges. In particular, the soliton energy per
topological charge (topological energy density) achieves a minimum neither for
nor high charges, but rather for intermediate values or
.Comment: 8 pages, 4 figure
Renormalizing Partial Differential Equations
In this review paper, we explain how to apply Renormalization Group ideas to
the analysis of the long-time asymptotics of solutions of partial differential
equations. We illustrate the method on several examples of nonlinear parabolic
equations. We discuss many applications, including the stability of profiles
and fronts in the Ginzburg-Landau equation, anomalous scaling laws in
reaction-diffusion equations, and the shape of a solution near a blow-up point.Comment: 34 pages, Latex; [email protected]; [email protected]
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