102 research outputs found
Hyper-complex four-manifolds from the Tzitz\'eica equation
It is shown how solutions to the Tzitz\'eica equation can be used to
construct a family of (pseudo) hyper-complex metrics in four dimensions.Comment: To be published in J.Math.Phy
Twistor theory of hyper-K{\"a}hler metrics with hidden symmetries
We briefly review the hierarchy for the hyper-K\"ahler equations and define a
notion of symmetry for solutions of this hierarchy. A four-dimensional
hyper-K\"ahler metric admits a hidden symmetry if it embeds into a hierarchy
with a symmetry. It is shown that a hyper-K\"ahler metric admits a hidden
symmetry if it admits a certain Killing spinor. We show that if the hidden
symmetry is tri-holomorphic, then this is equivalent to requiring symmetry
along a higher time and the hidden symmetry determines a `twistor group' action
as introduced by Bielawski \cite{B00}. This leads to a construction for the
solution to the hierarchy in terms of linear equations and variants of the
generalised Legendre transform for the hyper-K\"ahler metric itself given by
Ivanov & Rocek \cite{IR96}. We show that the ALE spaces are examples of
hyper-K\"ahler metrics admitting three tri-holomorphic Killing spinors. These
metrics are in this sense analogous to the 'finite gap' solutions in soliton
theory. Finally we extend the concept of a hierarchy from that of \cite{DM00}
for the four-dimensional hyper-K\"ahler equations to a generalisation of the
conformal anti-self-duality equations and briefly discuss hidden symmetries for
these equations.Comment: Final version. To appear in the August 2003 special issue of JMP on
`Integrability, Topological Solitons, and Beyond
Multidimensional integrable systems and deformations of Lie algebra homomorphisms
We use deformations of Lie algebra homomorphisms to construct deformations of
dispersionless integrable systems arising as symmetry reductions of
anti--self--dual Yang--Mills equations with a gauge group Diff.Comment: 14 pages. An example of a reduction to the Beltrami equation added.
New title. Final version, published in JM
Scalar--Flat Lorentzian Einstein--Weyl Spaces
We find all three-dimensional Einstein--Weyl spaces with the vanishing scalar
curvatureComment: 4 page
`Interpolating' differential reductions of multidimensional integrable hierarchies
We transfer the scheme of constructing differential reductions, developed
recently for the case of the Manakov-Santini hierarchy, to the general
multidimensional case. We consider in more detail the four-dimensional case,
connected with the second heavenly equation and its generalization proposed by
Dunajski. We give a characterization of differential reductions in terms of the
Lax-Sato equations as well as in the framework of the dressing method based on
nonlinear Riemann-Hilbert problem.Comment: Based on the talk at NLPVI, Gallipoli, 15 page
Solitons and admissible families of rational curves in twistor spaces
It is well known that twistor constructions can be used to analyse and to
obtain solutions to a wide class of integrable systems. In this article we
express the standard twistor constructions in terms of the concept of an
admissible family of rational curves in certain twistor spaces. Examples of of
such families can be obtained as subfamilies of a simple family of rational
curves using standard operations of algebraic geometry. By examination of
several examples, we give evidence that this construction is the basis of the
construction of many of the most important solitonic and algebraic solutions to
various integrable differential equations of mathematical physics. This is
presented as evidence for a principal that, in some sense, all soliton-like
solutions should be constructable in this way.Comment: 15 pages, Abstract and introduction rewritten to clarify the
objectives of the paper. This is the final version which will appear in
Nonlinearit
Interpolating Dispersionless Integrable System
We introduce a dispersionless integrable system which interpolates between
the dispersionless Kadomtsev-Petviashvili equation and the hyper-CR equation.
The interpolating system arises as a symmetry reduction of the anti--self--dual
Einstein equations in (2, 2) signature by a conformal Killing vector whose
self--dual derivative is null. It also arises as a special case of the
Manakov-Santini integrable system. We discuss the corresponding Einstein--Weyl
structures.Comment: 11 pages. New title, some errors corrected, section 5 removed. Final
version, to appear in J. Phys.
Strominger--Yau--Zaslow geometry, Affine Spheres and Painlev\'e III
We give a gauge invariant characterisation of the elliptic affine sphere
equation and the closely related Tzitz\'eica equation as reductions of real
forms of SL(3, \C) anti--self--dual Yang--Mills equations by two
translations, or equivalently as a special case of the Hitchin equation.
We use the Loftin--Yau--Zaslow construction to give an explicit expression
for a six--real dimensional semi--flat Calabi--Yau metric in terms of a
solution to the affine-sphere equation and show how a subclass of such metrics
arises from 3rd Painlev\'e transcendents.Comment: 38 pages. Final version. To appear in Communications in Mathematical
Physic
Solvable vector nonlinear Riemann problems, exact implicit solutions of dispersionless PDEs and wave breaking
We have recently solved the inverse spectral problem for integrable PDEs in
arbitrary dimensions arising as commutation of multidimensional vector fields
depending on a spectral parameter . The associated inverse problem, in
particular, can be formulated as a non linear Riemann Hilbert (NRH) problem on
a given contour of the complex plane. The most distinguished examples
of integrable PDEs of this type, like the dispersionless
Kadomtsev-Petviashivili (dKP), the heavenly and the 2 dimensional
dispersionless Toda equations, are real PDEs associated with Hamiltonian vector
fields. The corresponding NRH data satisfy suitable reality and symplectic
constraints. In this paper, generalizing the examples of solvable NRH problems
illustrated in \cite{MS4,MS5,MS6}, we present a general procedure to construct
solvable NRH problems for integrable real PDEs associated with Hamiltonian
vector fields, allowing one to construct implicit solutions of such PDEs
parametrized by an arbitrary number of real functions of a single variable.
Then we illustrate this theory on few distinguished examples for the dKP and
heavenly equations. For the dKP case, we characterize a class of similarity
solutions, a class of solutions constant on their parabolic wave front and
breaking simultaneously on it, and a class of localized solutions breaking in a
point of the plane. For the heavenly equation, we characterize two
classes of symmetry reductions.Comment: 29 page
A Comparison of the LVDP and {\Lambda}CDM Cosmological Models
We compare the cosmological kinematics obtained via our law of linearly
varying deceleration parameter (LVDP) with the kinematics obtained in the
{\Lambda}CDM model. We show that the LVDP model is almost indistinguishable
from the {\Lambda}CDM model up to the near future of our universe as far as the
current observations are concerned, though their predictions differ
tremendously into the far future.Comment: 6 pages, 5 figures, 1 table, matches the version to be published in
International Journal of Theoretical Physic
- …