123 research outputs found
Multi-Hamiltonian structure of Plebanski's second heavenly equation
We show that Plebanski's second heavenly equation, when written as a
first-order nonlinear evolutionary system, admits multi-Hamiltonian structure.
Therefore by Magri's theorem it is a completely integrable system. Thus it is
an example of a completely integrable system in four dimensions
Symmetry analysis and exact solutions of modified Brans-Dicke cosmological equations
We perform a symmetry analysis of modified Brans-Dicke cosmological equations
and present exact solutions. We discuss how the solutions may help to build
models of cosmology where, for the early universe, the expansion is linear and
the equation of state just changes the expansion velocity but not the
linearity. For the late universe the expansion is exponential and the effect of
the equation of state on the rate of expansion is just to change the constant
Hubble parameter.Comment: LaTeX2e source file, 14 pages, 7 reference
Anti-self-dual Riemannian metrics without Killing vectors, can they be realized on K3?
Explicit Riemannian metrics with Euclidean signature and anti-self dual
curvature that do not admit any Killing vectors are presented. The metric and
the Riemann curvature scalars are homogenous functions of degree zero in a
single real potential and its derivatives. The solution for the potential is a
sum of exponential functions which suggests that for the choice of a suitable
domain of coordinates and parameters it can be the metric on a compact
manifold. Then, by the theorem of Hitchin, it could be a class of metrics on
, or on surfaces whose universal covering is .Comment: Misprints in eqs.(9-11) corrected. Submitted to Classical and Quantum
Gravit
Solutions of the sDiff(2)Toda equation with SU(2) Symmetry
We present the general solution to the Plebanski equation for an H-space that
admits Killing vectors for an entire SU(2) of symmetries, which is therefore
also the general solution of the sDiff(2)Toda equation that allows these
symmetries. Desiring these solutions as a bridge toward the future for yet more
general solutions of the sDiff(2)Toda equation, we generalize the earlier work
of Olivier, on the Atiyah-Hitchin metric, and re-formulate work of Babich and
Korotkin, and Tod, on the Bianchi IX approach to a metric with an SU(2) of
symmetries. We also give careful delineations of the conformal transformations
required to ensure that a metric of Bianchi IX type has zero Ricci tensor, so
that it is a self-dual, vacuum solution of the complex-valued version of
Einstein's equations, as appropriate for the original Plebanski equation.Comment: 27 page
On the water-bag model of dispersionless KP hierarchy
We investigate the bi-Hamiltonian structure of the waterbag model of dKP for
two component case. One can establish the third-order and first-order
Hamiltonian operator associated with the waterbag model. Also, the dispersive
corrections are discussed.Comment: 19 page
Partner symmetries and non-invariant solutions of four-dimensional heavenly equations
We extend our method of partner symmetries to the hyperbolic complex
Monge-Amp\`ere equation and the second heavenly equation of Pleba\~nski. We
show the existence of partner symmetries and derive the relations between them
for both equations. For certain simple choices of partner symmetries the
resulting differential constraints together with the original heavenly
equations are transformed to systems of linear equations by an appropriate
Legendre transformation. The solutions of these linear equations are
generically non-invariant. As a consequence we obtain explicitly new classes of
heavenly metrics without Killing vectors.Comment: 20 pages, 1 table, corrected typo
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