86 research outputs found
A Note on Fractional KdV Hierarchies
We introduce a hierarchy of mutually commuting dynamical systems on a finite
number of Laurent series. This hierarchy can be seen as a prolongation of the
KP hierarchy, or a ``reduction'' in which the space coordinate is identified
with an arbitrarily chosen time of a bigger dynamical system. Fractional KdV
hierarchies are gotten by means of further reductions, obtained by constraining
the Laurent series. The case of sl(3)^2 and its bihamiltonian structure are
discussed in detail.Comment: Final version to appear in J. Math. Phys. Some changes in the order
of presentation, with more emphasis on the geometrical picture. One figure
added (using epsf.sty). 30 pages, Late
Spatially self-similar locally rotationally symmetric perfect fluid models
Einstein's field equations for spatially self-similar locally rotationally
symmetric perfect fluid models are investigated. The field equations are
rewritten as a first order system of autonomous ordinary differential
equations. Dimensionless variables are chosen in such a way that the number of
equations in the coupled system of differential equations is reduced as far as
possible. The system is subsequently analyzed qualitatively for some of the
models. The nature of the singularities occurring in the models is discussed.Comment: 27 pages, pictures available at
ftp://vanosf.physto.se/pub/figures/ssslrs.tar.g
Bianchi VIII Empty Futures
Using a qualitative analysis based on the Hamiltonian formalism and the
orthonormal frame representation we investigate whether the chaotic behaviour
which occurs close to the initial singularity is still present in the far
future of Bianchi VIII models. We describe some features of the vacuum Bianchi
VIII models at late times which might be relevant for studying the nature of
the future asymptote of the general vacuum inhomogeneous solution to the
Einstein field equations.Comment: 22 pages, no figures, Latex fil
Differential-difference equations associated with the fractional Lax operators
We study integrable hierarchies associated with spectral problems of the form
where are difference operators. The corresponding
nonlinear differential-difference equations can be viewed as inhomogeneous
generalizations of the Bogoyavlensky type lattices. While the latter turn into
the Korteweg--de Vries equation under the continuous limit, the lattices under
consideration provide discrete analogs of the Sawada--Kotera and
Kaup--Kupershmidt equations. The -matrix formulation and several simplest
explicit solutions are presented.Comment: 23 pages, 2 figure
Matter and dynamics in closed cosmologies
To systematically analyze the dynamical implications of the matter content in
cosmology, we generalize earlier dynamical systems approaches so that perfect
fluids with a general barotropic equation of state can be treated. We focus on
locally rotationally symmetric Bianchi type IX and Kantowski-Sachs orthogonal
perfect fluid models, since such models exhibit a particularly rich dynamical
structure and also illustrate typical features of more general cases. For these
models, we recast Einstein's field equations into a regular system on a compact
state space, which is the basis for our analysis. We prove that models expand
from a singularity and recollapse to a singularity when the perfect fluid
satisfies the strong energy condition. When the matter source admits Einstein's
static model, we present a comprehensive dynamical description, which includes
asymptotic behavior, of models in the neighborhood of the Einstein model; these
results make earlier claims about ``homoclinic phenomena and chaos'' highly
questionable. We also discuss aspects of the global asymptotic dynamics, in
particular, we give criteria for the collapse to a singularity, and we describe
when models expand forever to a state of infinite dilution; possible initial
and final states are analyzed. Numerical investigations complement the
analytical results.Comment: 23 pages, 24 figures (compressed), LaTe
Integrable boundary conditions for the Toda lattice
The problem of construction of the boundary conditions for the Toda lattice
compatible with its higher symmetries is considered. It is demonstrated that
this problem is reduced to finding of the differential constraints consistent
with the ZS-AKNS hierarchy. A method of their construction is offered based on
the B\"acklund transformations. It is shown that the generalized Toda lattices
corresponding to the non-exceptional Lie algebras of finite growth can be
obtained by imposing one of the four simplest integrable boundary conditions on
the both ends of the lattice. This fact allows, in particular, to solve the
problem of reduction of the series Toda lattices into the series ones.
Deformations of the found boundary conditions are presented which leads to the
Painlev\'e type equations.
Key words: Toda lattice, boundary conditions, integrability, B\"acklund
transformation, Lie algebras, Painlev\'e equation
Reduction and Realization in Toda and Volterra
We construct a new symplectic, bi-hamiltonian realization of the KM-system by
reducing the corresponding one for the Toda lattice. The bi-hamiltonian pair is
constructed using a reduction theorem of Fernandes and Vanhaecke. In this paper
we also review the important work of Moser on the Toda and KM-systems.Comment: 17 page
Differential-difference system related to toroidal Lie algebra
We present a novel differential-difference system in (2+1)-dimensional
space-time (one discrete, two continuum), arisen from the Bogoyavlensky's
(2+1)-dimensional KdV hierarchy. Our method is based on the bilinear identity
of the hierarchy, which is related to the vertex operator representation of the
toroidal Lie algebra \sl_2^{tor}.Comment: 10 pages, 4 figures, pLaTeX2e, uses amsmath, amssymb, amsthm,
graphic
On the Cosmology of Massive Vector Fields with SO(3) Global Symmetry
A relevant reference ([14]) has been added.Comment: 19 pages, plain tex, DF/IST-3/92 and DFFCUL 03-5/199
Timelike self-similar spherically symmetric perfect-fluid models
Einstein's field equations for timelike self-similar spherically symmetric
perfect-fluid models are investigated. The field equations are rewritten as a
first-order system of autonomous differential equations. Dimensionless
variables are chosen in such a way that the number of equations in the coupled
system is reduced as far as possible and so that the reduced phase space
becomes compact and regular. The system is subsequently analysed qualitatively
using the theory of dynamical systems.Comment: 23 pages, 6 eps-figure
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