57 research outputs found
A class of integrable lattices and KP hierarchy
We introduce a class of integrable -field first-order lattices together
with corresponding Lax equations. These lattices may be represented as
consistency condition for auxiliary linear systems defined on sequences of
formal dressing operators. This construction provides simple way to build
lattice Miura transformations between one-field lattice and -field () ones. We show that the lattices pertained to above class is in some sense
compatible with KP flows and define the chains of constrained KP Lax operators.Comment: LaTeX, 13 pages, accepted for publication in J. Phys. A: Math. Ge
Darboux Transformations, Infinitesimal Symmetries and Conservation Laws for Nonlocal Two-Dimensional Toda Lattice
The technique of Darboux transformation is applied to nonlocal partner of
two-dimensional periodic A_{n-1} Toda lattice. This system is shown to admit a
representation as the compatibility conditions of direct and dual
overdetermined linear systems with quantized spectral parameter. The
generalization of the Darboux transformation technique on linear equations of
such a kind is given. The connections between the solutions of overdetermined
linear systems and their expansions in series at singular points neighborhood
are presented. The solutions of the nonlocal Toda lattice and infinite
hierarchies of the infinitesimal symmetries and conservation laws are obtained.Comment: 12 pages, infinitesimal symmetries and conservation laws are adde
On some class of reductions for Itoh-Narita-Bogoyavlenskii lattice
We show a broad class of constraints compatible with
Itoh-Narita-Bogoyavlenskii lattice hierarchy. All these constraints can be
written in the form of discrete conservation law with appropriate
homogeneous polynomial discrete function .Comment: 15 page
Reductions of integrable lattices
Based on the notion of Darboux-KP chain hierarchy and its invariant
submanifolds we construct some class of constraints compatible with integrable
lattices. Some simple examples are given.Comment: 17 page
Kovalevski exponents and integrability properties in class A homogeneous cosmological models
Qualitative approach to homogeneous anisotropic Bianchi class A models in
terms of dynamical systems reveals a hierarchy of invariant manifolds. By
calculating the Kovalevski Exponents according to Adler - van Moerbecke method
we discuss how algebraic integrability property is distributed in this class of
models. In particular we find that algebraic nonintegrability of vacuum Bianchi
VII_0 model is inherited by more general Bianchi VIII and Bianchi IX vacuum
types. Matter terms (cosmological constant, dust and radiation) in the Einstein
equations typically generate irrational or complex Kovalevski exponents in
class A homogeneous models thus introducing an element of nonintegrability even
though the respective vacuum models are integrable.Comment: arxiv version is already officia
The Asymptotic Behaviour of Tilted Bianchi type VI Universes
We study the asymptotic behaviour of the Bianchi type VI universes with a
tilted -law perfect fluid. The late-time attractors are found for the
full 7-dimensional state space and for several interesting invariant subspaces.
In particular, it is found that for the particular value of the equation of
state parameter, , there exists a bifurcation line which signals a
transition of stability between a non-tilted equilibrium point to an extremely
tilted equilibrium point. The initial singular regime is also discussed and we
argue that the initial behaviour is chaotic for .Comment: 22 pages, 4 figures, to appear in CQ
A dynamical systems approach to the tilted Bianchi models of solvable type
We use a dynamical systems approach to analyse the tilting spatially
homogeneous Bianchi models of solvable type (e.g., types VI and VII)
with a perfect fluid and a linear barotropic -law equation of state. In
particular, we study the late-time behaviour of tilted Bianchi models, with an
emphasis on the existence of equilibrium points and their stability properties.
We briefly discuss the tilting Bianchi type V models and the late-time
asymptotic behaviour of irrotational Bianchi VII models. We prove the
important result that for non-inflationary Bianchi type VII models vacuum
plane-wave solutions are the only future attracting equilibrium points in the
Bianchi type VII invariant set. We then investigate the dynamics close to
the plane-wave solutions in more detail, and discover some new features that
arise in the dynamical behaviour of Bianchi cosmologies with the inclusion of
tilt. We point out that in a tiny open set of parameter space in the type IV
model (the loophole) there exists closed curves which act as attracting limit
cycles. More interestingly, in the Bianchi type VII models there is a
bifurcation in which a set of equilibrium points turn into closed orbits. There
is a region in which both sets of closed curves coexist, and it appears that
for the type VII models in this region the solution curves approach a
compact surface which is topologically a torus.Comment: 29 page
The Futures of Bianchi type VII0 cosmologies with vorticity
We use expansion-normalised variables to investigate the Bianchi type VII
model with a tilted -law perfect fluid. We emphasize the late-time
asymptotic dynamical behaviour of the models and determine their asymptotic
states. Unlike the other Bianchi models of solvable type, the type VII
state space is unbounded. Consequently we show that, for a general
non-inflationary perfect fluid, one of the curvature variables diverges at late
times, which implies that the type VII model is not asymptotically
self-similar to the future. Regarding the tilt velocity, we show that for
fluids with (which includes the important case of dust,
) the tilt velocity tends to zero at late times, while for a
radiation fluid, , the fluid is tilted and its vorticity is
dynamically significant at late times. For fluids stiffer than radiation
(), the future asymptotic state is an extremely tilted spacetime
with vorticity.Comment: 23 pages, v2:references and comments added, typos fixed, to appear in
CQ
The late-time behaviour of vortic Bianchi type VIII Universes
We use the dynamical systems approach to investigate the Bianchi type VIII
models with a tilted -law perfect fluid. We introduce
expansion-normalised variables and investigate the late-time asymptotic
behaviour of the models and determine the late-time asymptotic states. For the
Bianchi type VIII models the state space is unbounded and consequently, for all
non-inflationary perfect fluids, one of the curvature variables grows without
bound. Moreover, we show that for fluids stiffer than dust (), the
fluid will in general tend towards a state of extreme tilt. For dust
(), or for fluids less stiff than dust (), we show that
the fluid will in the future be asymptotically non-tilted. Furthermore, we show
that for all the universe evolves towards a vacuum state but
does so rather slowly, .Comment: 19 pages, 3 ps figures, v2:typos fixed, refs and more discussion
adde
The mixmaster universe: A chaotic Farey tale
When gravitational fields are at their strongest, the evolution of spacetime
is thought to be highly erratic. Over the past decade debate has raged over
whether this evolution can be classified as chaotic. The debate has centered on
the homogeneous but anisotropic mixmaster universe. A definite resolution has
been lacking as the techniques used to study the mixmaster dynamics yield
observer dependent answers. Here we resolve the conflict by using observer
independent, fractal methods. We prove the mixmaster universe is chaotic by
exposing the fractal strange repellor that characterizes the dynamics. The
repellor is laid bare in both the 6-dimensional minisuperspace of the full
Einstein equations, and in a 2-dimensional discretisation of the dynamics. The
chaos is encoded in a special set of numbers that form the irrational Farey
tree. We quantify the chaos by calculating the strange repellor's Lyapunov
dimension, topological entropy and multifractal dimensions. As all of these
quantities are coordinate, or gauge independent, there is no longer any
ambiguity--the mixmaster universe is indeed chaotic.Comment: 45 pages, RevTeX, 19 Figures included, submitted to PR
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