171 research outputs found
The initial singularity of ultrastiff perfect fluid spacetimes without symmetries
We consider the Einstein equations coupled to an ultrastiff perfect fluid and
prove the existence of a family of solutions with an initial singularity whose
structure is that of explicit isotropic models. This family of solutions is
`generic' in the sense that it depends on as many free functions as a general
solution, i.e., without imposing any symmetry assumptions, of the
Einstein-Euler equations. The method we use is a that of a Fuchsian reduction.Comment: 16 pages, journal versio
Fuchsian analysis of singularities in Gowdy spacetimes beyond analyticity
Fuchsian equations provide a way of constructing large classes of spacetimes
whose singularities can be described in detail. In some of the applications of
this technique only the analytic case could be handled up to now. This paper
develops a method of removing the undesirable hypothesis of analyticity. This
is applied to the specific case of the Gowdy spacetimes in order to show that
analogues of the results known in the analytic case hold in the smooth case. As
far as possible the likely strengths and weaknesses of the method as applied to
more general problems are displayed.Comment: 14 page
Psi-series solutions of the cubic H\'{e}non-Heiles system and their convergence
The cubic H\'enon-Heiles system contains parameters, for most values of
which, the system is not integrable. In such parameter regimes, the general
solution is expressible in formal expansions about arbitrary movable branch
points, the so-called psi-series expansions. In this paper, the convergence of
known, as well as new, psi-series solutions on real time intervals is proved,
thereby establishing that the formal solutions are actual solutions
Quasilinear hyperbolic Fuchsian systems and AVTD behavior in T2-symmetric vacuum spacetimes
We set up the singular initial value problem for quasilinear hyperbolic
Fuchsian systems of first order and establish an existence and uniqueness
theory for this problem with smooth data and smooth coefficients (and with even
lower regularity). We apply this theory in order to show the existence of
smooth (generally not analytic) T2-symmetric solutions to the vacuum Einstein
equations, which exhibit AVTD (asymptotically velocity term dominated) behavior
in the neighborhood of their singularities and are polarized or half-polarized.Comment: 78 page
Oscillons in dilaton-scalar theories
It is shown by both analytical methods and numerical simulations that
extremely long living spherically symmetric oscillons appear in virtually any
real scalar field theory coupled to a massless dilaton (DS theories). In fact
such "dilatonic" oscillons are already present in the simplest non-trivial DS
theory -- a free massive scalar field coupled to the dilaton. It is shown that
in analogy to the previously considered cases with a single nonlinear scalar
field, in DS theories there are also time periodic quasibreathers (QB)
associated to small amplitude oscillons. Exploiting the QB picture the
radiation law of the small amplitude dilatonic oscillons is determined
analytically.Comment: extended discussion on stability, to appear in JHEP, 29 pages, 7
figure
Low regularity solutions of two fifth-order KdV type equations
The Kawahara and modified Kawahara equations are fifth-order KdV type
equations and have been derived to model many physical phenomena such as
gravity-capillary waves and magneto-sound propagation in plasmas. This paper
establishes the local well-posedness of the initial-value problem for Kawahara
equation in with and the local well-posedness
for the modified Kawahara equation in with .
To prove these results, we derive a fundamental estimate on dyadic blocks for
the Kawahara equation through the multiplier norm method of Tao
\cite{Tao2001} and use this to obtain new bilinear and trilinear estimates in
suitable Bourgain spaces.Comment: 17page
Locally U(1)*U(1) Symmetric Cosmological Models: Topology and Dynamics
We show examples which reveal influences of spatial topologies to dynamics,
using a class of spatially {\it closed} inhomogeneous cosmological models. The
models, called the {\it locally U(1)U(1) symmetric models} (or the {\it
generalized Gowdy models}), are characterized by the existence of two commuting
spatial {\it local} Killing vectors. For systematic investigations we first
present a classification of possible spatial topologies in this class. We
stress the significance of the locally homogeneous limits (i.e., the Bianchi
types or the `geometric structures') of the models. In particular, we show a
method of reduction to the natural reduced manifold, and analyze the
equivalences at the reduced level of the models as dynamical models. Based on
these fundamentals, we examine the influence of spatial topologies on dynamics
by obtaining translation and reflection operators which commute with the
dynamical flow in the phase space.Comment: 32 pages, 1 figure, LaTeX2e, revised Introduction slightly. To appear
in CQ
The Gowdy T3 Cosmologies revisited
We have examined, repeated and extended earlier numerical calculations of
Berger and Moncrief for the evolution of unpolarized Gowdy T3 cosmological
models. Our results are consistent with theirs and we support their claim that
the models exhibit AVTD behaviour, even though spatial derivatives cannot be
neglected. The behaviour of the curvature invariants and the formation of
structure through evolution both backwards and forwards in time is discussed.Comment: 11 pages, LaTeX, 6 figures, results and conclusions revised and
(considerably) expande
Analytic and Asymptotic Methods for Nonlinear Singularity Analysis: a Review and Extensions of Tests for the Painlev\'e Property
The integrability (solvability via an associated single-valued linear
problem) of a differential equation is closely related to the singularity
structure of its solutions. In particular, there is strong evidence that all
integrable equations have the Painlev\'e property, that is, all solutions are
single-valued around all movable singularities. In this expository article, we
review methods for analysing such singularity structure. In particular, we
describe well known techniques of nonlinear regular-singular-type analysis,
i.e. the Painlev\'e tests for ordinary and partial differential equations. Then
we discuss methods of obtaining sufficiency conditions for the Painlev\'e
property. Recently, extensions of \textit{irregular} singularity analysis to
nonlinear equations have been achieved. Also, new asymptotic limits of
differential equations preserving the Painlev\'e property have been found. We
discuss these also.Comment: 40 pages in LaTeX2e. To appear in the Proceedings of the CIMPA Summer
School on "Nonlinear Systems," Pondicherry, India, January 1996, (eds) B.
Grammaticos and K. Tamizhman
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