arXiv subjects

Y. Nutku

Publications and source records attributed to Y. Nutku.

24 records · Page 2Linked to original sources

arXiv1998

Hamiltonian structure of real Monge-Ampère equations

The real homogeneous Monge-Ampère 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-Ampère 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-Ampère 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.

solv-int↗

arXiv1998

Harmonic Map Formulation of Colliding Electrovac Plane Waves

The formulation of the Einstein field equations admitting two Killing vectors in terms of harmonic mappings of Riemannian manifolds is a subject in which Charlie Misner has played a pioneering role. We shall consider the hyperbolic case of the Einstein-Maxwell equations admitting two hypersurface orthogonal Killing vectors which physically describes the interaction of two electrovac plane waves. Following Penrose's discussion of the Cauchy problem we shall present the initial data appropriate to this collision problem. We shall also present three different ways in which the Einstein-Maxwell equations for colliding plane wave spacetimes can be recognized as a harmonic map. The goal is to cast the Einstein-Maxwell equations into a form adopted to the initial data for colliding impulsive gravitational and electromagnetic shock waves in such a way that a simple harmonic map will directly yield the metric and the Maxwell potential 1-form of physical interest. For Charles W. Misner on his 60th birthday

gr-qc↗

arXiv1998

Gravitational Instantons from Minimal Surfaces

Physical properties of gravitational instantons which are derivable from minimal surfaces in 3-dimensional Euclidean space are examined using the Newman-Penrose formalism for Euclidean signature. The gravitational instanton that corresponds to the helicoid minimal surface is investigated in detail. This is a metric of Bianchi Type $VII_0$, or E(2) which admits a hidden symmetry due to the existence of a quadratic Killing tensor. It leads to a complete separation of variables in the Hamilton-Jacobi equation for geodesics, as well as in Laplace's equation for a massless scalar field. The scalar Green function can be obtained in closed form which enables us to calculate the vacuum fluctuations of a massless scalar field in the background of this instanton.

gr-qc↗

arXiv1998

Gravitational instantons admit hyper-Kähler structure

We construct the explicit form of three almost complex structures that a Riemannian manifold with self-dual curvature admits and show that their Nijenhuis tensors vanish so that they are integrable. This proves that gravitational instantons with self-dual curvature admit hyper-Kähler structure. In order to arrive at the three vector valued 1-forms defining almost complex structure, we give a spinor description of real 4-dimensional Riemanian manifolds with Euclidean signature in terms of two independent sets of 2-component spinors. This is a version of the original Newman-Penrose formalism that is appropriate to the discussion of the mathematical, as well as physical properties of gravitational instantons. We shall build on the work of Goldblatt who first developed an NP formalism for gravitational instantons but we shall adopt it to differential forms in the NP basis to make the formalism much more compact. We shall show that the spin coefficients, connection 1-form, curvature 2-form, Ricci and Bianchi identities, as well as the Maxwell equations naturally split up into their self-dual and anti-self-dual parts corresponding to the two independent spin frames. We shall give the complex dyad as well as the spinor formulation of the almost complex structures and show that they reappear under the guise of a triad basis for the Petrov classification of gravitational instantons. Completing the work of Salamon on hyper-Kähler structure, we show that the vanishing of the Nijenhuis tensor for all three almost complex structures depends on the choice of a self-dual gauge for the connection which is guaranteed by virtue of the fact that the curvature 2-form is self-dual for gravitational instantons.

gr-qc↗

arXiv1998

Alternative bi-Hamiltonian structures for WDVV equations of associativity

The WDVV equations of associativity in 2-d topological field theory are completely integrable third order Monge-Ampère equations which admit bi-Hamiltonian structure. The time variable plays a distinguished role in the discussion of Hamiltonian structure whereas in the theory of WDVV equations none of the independent variables merits such a distinction. WDVV equations admit very different alternative Hamiltonian structures under different possible choices of the time variable but all these various Hamiltonian formulations can be brought together in the framework of the covariant theory of symplectic structure. They can be identified as different components of the covariant Witten-Zuckerman symplectic 2-form current density where a variational formulation of the WDVV equation that leads to the Hamiltonian operator through the Dirac bracket is available.

hep-th↗

arXiv1994

On the Monge-Ampere equivalent of the sine-Gordon equation

Surfaces of constant negative curvature in Euclidean space can be described by either the sine-Gordon equation for the angle between asymptotic directions, or a Monge-Ampere equation for the graph of the surface. We present the explicit form of the correspondence between these two integrable non-linear partial differential equations using their well-known properties in differential geometry. We find that the cotangent of the angle between asymptotic directions is directly related to the mean curvature of the surface. This is a Backlund-type transformation between the sine-Gordon and Monge-Ampere equations.

solv-int↗