arXiv subjects

Peter Leach

Publications and source records attributed to Peter Leach.

9 recordsLinked to original sources

arXiv2001

Symmetry, singularities and intregrability in complex dynamics VII: Integrability Properties of FRW-Scalar Cosmologies

This paper considers some physically interesting cosmological dynamical systems in the FRW-scalarfield category which are examined for integrability according to the criterion of Painlevé. In the literature these systems have been examined from the point of view of dynamical systems and the results from the two disparate methods of analysis are compared. This allows some more general comments to be made on the use of the Painlevé method in covariant systems.

gr-qc↗

arXiv2001

First integrals and parametric solutions for equations integrable through Lie symmetries

We present here the explicit parametric solutions of second order differential equations invariant under time translation and rescaling and third order differential equations invariant under time translation and the two homogeneity symmetries. The computation of first integrals gives in the most general case, the parametric form of the general solution. For some polynomial functions we obtain a time parametrisation quadrature which can be solved in terms of ``known'' functions.

nlin.SI↗

arXiv2001

Is Nature Generic?

An introductory guide to mathematical cosmology is given focusing on the issue of the genericity of various important results which have been obtained during the last thirty or so years. Some of the unsolved problems along with certain new and potentially powerful methods which may be used for future progress are also given from a unified perspective.

gr-qc↗

arXiv2001

Symmetry, singularities and integrability in complex dynamics III: approximate symmetries and invariants

The different natures of approximate symmetries and their corresponding first integrals/invariants are delineated in the contexts of both Lie symmetries of ordinary differential equations and Noether symmetries of the Action Integral. Particular note is taken of the effect of taking higher orders of the perturbation parameter. Approximate symmetries of approximate first integrals/invariants and the problems of calculating them using the Lie method are considered.

nlin.SI↗

arXiv2000

Symmetries of homogeneous cosmologies

We reformulate the dynamics of homogeneous cosmologies with a scalar field matter source with an arbitrary self-interaction potential in the language of jet bundles and extensions of vector fields. In this framework, the Bianchi-scalar field equations become subsets of the second Bianchi jet bundle, $J^2$, and every Bianchi cosmology is naturally extended to live on a variety of $J^2$. We are interested in the existence and behaviour of extensions of arbitrary Bianchi-Lie and variational vector fields acting on the Bianchi variety and accordingly we classify all such vector fields corresponding to both Bianchi classes $A$ and $B$. We give examples of functions defined on Bianchi jet bundles which are constant along some Bianchi models (first integrals) and use these to find particular solutions in the Bianchi total space. We discuss how our approach could be used to shed new light to questions like isotropization and the nature of singularities of homogeneous cosmologies by examining the behaviour of the variational vector fields and also give rise to interesting questions about the `evolution' and nature of the cosmological symmetries themselves.

gr-qc↗

arXiv2000

Symmetry, Singularities and Integrability in Complex Dynamics IV: Painleve Integrability of Isotropic Cosmologies

We apply the results of singularity analysis to the isotropic cosmological models in general relativity and string theory with a variety of matter terms. For some of these models the standard Painlevé test is sufficient to demonstrate integrability or nonintegrability in the sense of Painlevé. For others of these models it is necessary to use a less algorithmic procedure.

gr-qc↗

arXiv2000

Ever-expanding, Isotropizing, Quadratic Cosmologies

We consider some aspects of the global evolution problem of Hamiltonian homogeneous, anisotropic cosmologies derived from a purely quadratic action functional of the scalar curvature. We show that models can isotropize in the positive asymptotic direction and that quadratic diagonal Bianchi IX models do not recollapse and may be regular initially. Although the global existence and isotropization results we prove hold quite generally, they are applied to specific Bianchi models in an attempt to describe how certain dynamical properties uncommon to the general relativity case, become generic features of these quadratic universes. The question of integrability of the models is also considered. Our results point to the fact that the more general models are not integrable in the sense of Painlevé and for the Bianchi IX case this may be connected to the validity of a BKL oscillatory picture on approach to the singularity in sharp contrast with other higher order gravity theories that contain an Einstein term and show a monotonic evolution towards the initial singularity.

gr-qc↗

arXiv2000

Symmetry, singularities and integrability in complex dynamics II: Rescaling and time-translation in two-dimensional systems

The explicit integrability of second order ordinary differential equations invariant under time-translation and rescaling is investigated. Quadratic systems generated from the linearisable version of this class of equations are analysed to determine therelationship between the Painlevé and singularity properties of the different systems. The transformation contains a parameter and for critical values, intimately related to the possession of the Painlevé Property in the parent second order equation, one finds a difference from the generic behaviour. This study is a prelude to a full discussion of the class of transformations which preserve the Painlevé Property in the construction of quadratic systems from scalar nonlinear odes invariant under time translation and rescaling.

nlin.SI↗

arXiv2000

Symmetry, singularities and integrability in complex dynamics I: the reduction problem

Quadratic systems generated using Yang-Baxter equations are integrable in a sense, but we display a deterioration in the possession of the Painlevé property as the number of equations in each `integrable system' increases. Certain intermediate systems are constructed and also tested for the Painlevé property. The Lie symmetries are also computed for completeness.

nlin.SI↗