12 research outputs found
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
Integrable structures for a generalized Monge-AmpĂšre equation
We consider a 3rd-order generalized Monge-AmpeÌre equa-
tion u yyy â u 2 xxy + u xxx u xyy = 0 (which is closely related to the asso-
ciativity equation in the 2-d topological field theory) and describe all
integrable structures related to it (i.e., Hamiltonian, symplectic, and re-
cursion operators). Infinite hierarchies of symmetries and conservation
laws are constructed as well