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 $P\psi=\lambda Q\psi$ where $P,Q$ 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 $r$-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 $A$ Toda lattices into the series $D$ 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