623 research outputs found
Superposition Formulas for Darboux Integrable Exterior Differential Systems
In this paper we present a far-reaching generalization of E. Vessiot's
analysis of the Darboux integrable partial differential equations in one
dependent and two independent variables. Our approach provides new insights
into this classical method, uncovers the fundamental geometric invariants of
Darboux integrable systems, and provides for systematic, algorithmic
integration of such systems. This work is formulated within the general
framework of Pfaffian exterior differential systems and, as such, has
applications well beyond those currently found in the literature. In
particular, our integration method is applicable to systems of hyperbolic PDE
such as the Toda lattice equations, 2 dimensional wave maps and systems of
overdetermined PDE.Comment: 80 page report. Updated version with some new sections, and major
improvements to other
Backlund Transformations for Darboux Integrable Differential Systems
We give a new mechanism for constructing Backlund transformations by using
symmetry reduction of differential systems. We then characterize a family of
Backlund transformations between Darboux integrable systems where the Backlund
transformation can be constructed by the proposed symmetry reduction method.Comment: 61 page
Control of clover infertility in sheep
A summary of practices recommended for the control of infertility caused by subterranean clover in West Australian sheep.
PROLONGED grazing of green subterranean clover pastures often reduces ewe fertility. In more extreme cases, obvious signs of clover disease occur
Group Invariant Solutions Without Transversality
We present a generalization of Lie\u27s method for finding the group invariant solutions to a system of partial differential equations. Our generalization relaxes the standard transversality assumption and encompasses the common situation where the reduced differential equations for the group invariant solutions involve both fewer dependent and independent variables. The theoretical basis for our method is provided by a general existence theorem for the invariant sections, both local and global, of a bundle on which a finite dimensional Lie group acts. A simple and natural extension of our characterization of invariant sections leads to an intrinsic characterization of the reduced equations for the group invariant solutions for a system of differential equations. The characterization of both the invariant sections and the reduced equations are summarized schematically by the kinematic and dynamic reduction diagrams and are illustrated by a number of examples from fluid mechanics, harmonic maps, and general relativity. This work also provides the theoretical foundations for a further detailed study of the reduced equations for group invariant solutions
Shortcuts to high symmetry solutions in gravitational theories
We apply the Weyl method, as sanctioned by Palais' symmetric criticality
theorems, to obtain those -highly symmetric -geometries amenable to explicit
solution, in generic gravitational models and dimension. The technique consists
of judiciously violating the rules of variational principles by inserting
highly symmetric, and seemingly gauge fixed, metrics into the action, then
varying it directly to arrive at a small number of transparent, indexless,
field equations. Illustrations include spherically and axially symmetric
solutions in a wide range of models beyond D=4 Einstein theory; already at D=4,
novel results emerge such as exclusion of Schwarzschild solutions in cubic
curvature models and restrictions on ``independent'' integration parameters in
quadratic ones. Another application of Weyl's method is an easy derivation of
Birkhoff's theorem in systems with only tensor modes. Other uses are also
suggested.Comment: 10 page
Transverse Group Actions on Bundles
An action of a Lie group G on a bundle is said to be transverse if it is projectable and if the orbits of G on E are diffeomorphic under π to the orbits of G on M. Transverse group actions on bundles are completely classified in terms of the pullback bundle construction for G-invariant maps. This classification result is used to give a full characterization of the G invariant sections of E for projectable group actions
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