24 research outputs found
Iterated Differential Forms II: Riemannian Geometry Revisited
A natural extension of Riemannian geometry to a much wider context is
presented on the basis of the iterated differential form formalism developed in
math.DG/0605113 and an application to general relativity is given.Comment: 12 pages, extended version of the published note Dokl. Math. 73, n. 2
(2006) 18
A unified approach to computation of integrable structures
We expose (without proofs) a unified computational approach to integrable
structures (including recursion, Hamiltonian, and symplectic operators) based
on geometrical theory of partial differential equations. We adopt a coordinate
based approach and aim to provide a tutorial to the computations.Comment: 19 pages, based on a talk on the SPT 2011 conference,
http://www.sptspt.it/spt2011/ ; v2, v3: minor correction
Algebraic properties of Gardner's deformations for integrable systems
An algebraic definition of Gardner's deformations for completely integrable
bi-Hamiltonian evolutionary systems is formulated. The proposed approach
extends the class of deformable equations and yields new integrable
evolutionary and hyperbolic Liouville-type systems. An exactly solvable
two-component extension of the Liouville equation is found.Comment: Proc. conf. "Nonlinear Physics: Theory and Experiment IV" (Gallipoli,
2006); Theor. Math. Phys. (2007) 151:3/152:1-2, 16p. (to appear
Symmetry classification of third-order nonlinear evolution equations. Part I: Semi-simple algebras
We give a complete point-symmetry classification of all third-order evolution
equations of the form
which admit semi-simple symmetry algebras and extensions of these semi-simple
Lie algebras by solvable Lie algebras. The methods we employ are extensions and
refinements of previous techniques which have been used in such
classifications.Comment: 53 page
Lower-order ODEs to determine new twisting type N Einstein spaces via CR geometry
In the search for vacuum solutions, with or without a cosmological constant,
of the Einstein field equations of Petrov type N with twisting principal null
directions, the CR structures to describe the parameter space for a congruence
of such null vectors provide a very useful tool. A work of Hill, Lewandowski
and Nurowski has given a good foundation for this, reducing the field equations
to a set of differential equations for two functions, one real, one complex, of
three variables. Under the assumption of the existence of one Killing vector,
the (infinite-dimensional) classical symmetries of those equations are
determined and group-invariant solutions are considered. This results in a
single ODE of the third order which may easily be reduced to one of the second
order. A one-parameter class of power series solutions, g(w), of this
second-order equation is realized, holomorphic in a neighborhood of the origin
and behaving asymptotically as a simple quadratic function plus lower-order
terms for large values of w, which constitutes new solutions of the twisting
type N problem. The solution found by Leroy, and also by Nurowski, is shown to
be a special case in this class. Cartan's method for determining equivalence of
CR manifolds is used to show that this class is indeed much more general.
In addition, for a special choice of a parameter, this ODE may be integrated
once, to provide a first-order Abel equation. It can also determine new
solutions to the field equations although no general solution has yet been
found for it.Comment: 28 page