Yavuz Nutku

Hamiltonian structures for systems of hyperbolic conservation laws

The bi‐Hamiltonian structure for a large class of one‐dimensional hyberbolic systems of conservation laws in two field variables, including the equations of gas dynamics, shallow water waves, one‐dimensional elastic media, and the Born–Infeld equation from nonlinear electrodynamics, is exhibited. For polytropic gas dynamics, these results lead to a quadri‐Hamiltonian structure. New higher‐order entropy‐flux pairs (conservation laws) and higher‐order symmetries are exhibited.

On a new class of completely integrable nonlinear wave equations. II. Multi‐Hamiltonian structure

The multi‐Hamiltonian structure of a class of nonlinear wave equations governing the propagation of finite amplitude waves is discussed. Infinitely many conservation laws had earlier been obtained for these equations. Starting from a (primary) Hamiltonian formulation of these equations the necessary and sufficient conditions for the existence of bi‐Hamiltonian structure are obtained and it is shown that the second Hamiltonian operator can be constructed solely through a knowledge of the first Hamiltonian function. The recursion operator which first appears at the level of bi‐Hamiltonian structure gives rise to an infinite sequence of conserved Hamiltonians. It is found that in general there exist two different infinite sequences of conserved quantities for these equations. The recursion relation defining higher Hamiltonian structures enables one to obtain the necessary and sufficient conditions for the existence of the (k+1)st Hamiltonian operator which depends on the kth Hamiltonian function. The infinite sequence of conserved Hamiltonians are common to all the higher Hamiltonian structures. The equations of gas dynamics are discussed as an illustration of this formalism and it is shown that in general they admit tri‐Hamiltonian structure with two distinct infinite sets of conserved quantities. The isothermal case of γ=1 is an exceptional one that requires separate treatment. This corresponds to a specialization of the equations governing the expansion of plasma into vacuum which will be shown to be equivalent to Poisson’s equation in nonlinear acoustics.

Multi-Lagrangians for integrable systems

We propose a general scheme to construct multiple Lagrangians for completely integrable nonlinear evolution equations that admit multi-Hamiltonian structure. The recursion operator plays a fundamental role in this construction. We use a conserved quantity higher/lower than the Hamiltonian in the potential part of the new Lagrangian and determine the corresponding kinetic terms by generating the appropriate momentum map. This leads to some remarkable new developments. We show that nonlinear evolutionary systems that admit N-fold first order local Hamiltonian structure can be cast into variational form with 2N−1 Lagrangians which will be local functionals of Clebsch potentials. This number increases to 3N−2 when the Miura transformation is invertible. Furthermore we construct a new Lagrangian for polytropic gas dynamics in 1+1 dimensions which is a free, local functional of the physical field variables, namely density and velocity, thus dispensing with the necessity of introducing Clebsch potentials entirel...

Multi-Hamiltonian structure of Plebanski's second heavenly equation

We show that Plebanskis second heavenly equation, when written as a first-order nonlinear evolutionary system, admits multi-Hamiltonian structure. Therefore by Magris theorem it is a completely integrable system. Thus it is an example of a completely integrable system in four dimensions.

Partner symmetries and non-invariant solutions of four-dimensional heavenly equations

We extend our method of partner symmetries to the hyperbolic complex Monge-Ampere equation and the second heavenly equation of Plebanski. We show the existence of partner symmetries and derive the relations between them. 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.

Hamiltonian structure of real Monge - Ampère equations

The variational principle for the real homogeneous Monge - Ampere equation in two dimensions is shown to contain three arbitrary functions of four variables. There exist two different specializations of this variational principle where the Lagrangian is degenerate and furthermore contains an arbitrary function of two variables. The Hamiltonian formulation of these degenerate Lagrangian systems requires the use of Diracs theory of constraints. As in the case of most completely integrable systems the constraints are second class and Dirac brackets directly yield the Hamiltonian operators. Thus the real homogeneous Monge - Ampere equation in two dimensions admits two classes of infinitely many Hamiltonian operators, namely a family of local, as well as another family non-local Hamiltonian operators and symplectic 2-forms which depend on arbitrary functions of two variables. The simplest non-local Hamiltonian operator corresponds to the Kac - Moody algebra of vector fields and functions on the unit circle. Hamiltonian operators that belong to either class are compatible with each other but between classes there is only one compatible pair. In the case of real Monge - Ampere equations with constant right-hand side this compatible pair is the only pair of Hamiltonian operators that survives. Then the complete integrability of all these real Monge - Ampere equations follows by Magris theorem. Some of the remarkable properties we have obtained for the Hamiltonian structure of the real homogeneous Monge - Ampere equation in two dimensions turn out to be generic to the real homogeneous Monge - Ampere equation and the geodesic flow for the complex homogeneous Monge - Ampere equation in arbitrary number of dimensions. Hence among all integrable nonlinear evolution equations in one space and one time dimension, the real homogeneous Monge - Ampere equation is distinguished as one that retains its character as an integrable system in multiple dimensions.

New nonlinear evolution equations from surface theory

We point out that the connection between surfaces in three‐dimensional flat space and the inverse scattering problem provides a systematic way for constructing new nonlinear evolution equations. In particular we study the imbedding for Guichard surfaces which gives rise to the Calapso–Guichard equations generalizing the sine‐Gordon (SG) equation. Further, we investigate the geometry of surfaces and their imbedding which results in the Korteweg–deVries (KdV) equation. Then by constructing a family of applicable surfaces we obtain a generalization of the KdV equation to a compressible fluid.

Gravitational instantons from minimal surfaces

Physical properties of gravitational instantons which are derivable from minimal surfaces in three-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 , 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 Laplaces 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.

A family of heavenly metrics

This is a corrected and essentially extended version of the unpublished manuscript by Y Nutku and M Sheftel which contains new results. It is proposed to be published in honour of Y Nutku’s memory. All corrections and new results in sections 1, 2 and 4 are due to M Sheftel. We present new anti-self-dual exact solutions of the Einstein field equations with Euclidean and neutral (ultra-hyperbolic) signatures that admit only one rotational Killing vector. Such solutions of the Einstein field equations are determined by non-invariant solutions of Boyer–Finley (BF) equation. For the case of Euclidean signature such a solution of the BF equation was first constructed by Calderbank and Tod. Two years later, Martina, Sheftel and Winternitz applied the method of group foliation to the BF equation and reproduced the Calderbank–Tod solution together with new solutions for the neutral signature. In the case of Euclidean signature we obtain new metrics which asymptotically locally look like a flat space and have a non-removable singular point at the origin. In the case of ultra-hyperbolic signature there exist three inequivalent forms of metric. Only one of these can be obtained by analytic continuation from the Calderbank–Tod solution whereas the other two are new.

Quantization with maximally degenerate Poisson brackets: the harmonic oscillator!

Nambus construction of multi-linear brackets for super-integrable systems can be thought of as degenerate Poisson brackets with a maximal set of Casimirs in their kernel. By introducing privileged coordinates in phase space these degenerate Poisson brackets are brought to the form of Heisenbergs equations. We propose a definition for constructing quantum operators for classical functions, which enables us to turn the maximally degenerate Poisson brackets into operators. They pose a set of eigenvalue problems for a new state vector. The requirement of the single-valuedness of this eigenfunction leads to quantization. The example of the harmonic oscillator is used to illustrate this general procedure for quantizing a class of maximally super-integrable systems.