18 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
Partner symmetries of the complex Monge-Ampere equation yield hyper-Kahler metrics without continuous symmetries
We extend the Mason-Newman Lax pair for the elliptic complex Monge-Amp\`ere
equation so that this equation itself emerges as an algebraic consequence. We
regard the function in the extended Lax equations as a complex potential. We
identify the real and imaginary parts of the potential, which we call partner
symmetries, with the translational and dilatational symmetry characteristics
respectively. Then we choose the dilatational symmetry characteristic as the
new unknown replacing the K\"ahler potential which directly leads to a Legendre
transformation and to a set of linear equations satisfied by a single real
potential. This enables us to construct non-invariant solutions of the Legendre
transform of the complex Monge-Amp\`ere equation and obtain hyper-K\"ahler
metrics with anti-self-dual Riemann curvature 2-form that admit no Killing
vectors.Comment: submitted to J. Phys.
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
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
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
Hamiltonian structure of real Monge-Amp\`ere equations
The real homogeneous Monge-Amp\`{e}re equation in one space and one time
dimensions admits infinitely many Hamiltonian operators and is completely
integrable by Magri's theorem. This remarkable property holds in arbitrary
number of dimensions as well, so that among all integrable nonlinear evolution
equations the real homogeneous Monge-Amp\`{e}re equation is distinguished as
one that retains its character as an integrable system in multi-dimensions.
This property can be traced back to the appearance of arbitrary functions in
the Lagrangian formulation of the real homogeneous Monge-Amp\`ere equation
which is degenerate and requires use of Dirac's theory of constraints for its
Hamiltonian formulation. As in the case of most completely integrable systems
the constraints are second class and Dirac brackets directly yield the
Hamiltonian operators. The simplest Hamiltonian operator results in the
Kac-Moody algebra of vector fields and functions on the unit circle.Comment: published in J. Phys. A 29 (1996) 325