Implicit Solutions of PDE's

Further investigations of implicit solutions to non-linear partial differential equations are pursued. Of particular interest are the equations which are Lorentz invariant. The question of which differential equations of second order for a single unknown $\phi$ are solved by the imposition of an inhomogeneous quadratic relationship among the independent variables, whose coefficients are functions of $\phi$ is discussed, and it is shown that if the discriminant of the quadratic vanishes, then an implicit solution of the so-called Universal Field Equation is obtained. The relation to the general solution is discussed.Comment: 11 pages LaTeX2

Moyal Brackets in M-Theory

The infinite limit of Matrix Theory in 4 and 10 dimensions is described in terms of Moyal Brackets. In those dimensions there exists a Bogomol'nyi bound to the Euclideanized version of these equations, which guarantees that solutions of the first order equations also solve the second order Matrix Theory equations. A general construction of such solutions in terms of a representation of the target space co-ordinates as non-local spinor bilinears, which are generalisations of the standard Wigner functions on phase space, is given.Comment: 10 pages, Latex, no figures. References altered, typos correcte

Integrable Top Equations associated with Projective Geometry over Z_2

We give a series of integrable top equations associated with the projective geometry over Z_2 as a (2^n-1)-dimensional generalisation of the 3D Euler top equations. The general solution of the (2^n-1)D top is shown to be given by an integration over a Riemann surface with genus (2^{n-1}-1)^2.Comment: 8 pages, Late

Linearisation of Universal Field Equations

The Universal Field Equations, recently constructed as examples of higher dimensional dynamical systems which admit an infinity of inequivalent Lagrangians are shown to be linearised by a Legendre transformation. This establishes the conjecture that these equations describe integrable systems. While this construction is implicit in general, there exists a large class of solutions for which an explicit form may be written.Comment: 11pp., DTP-92/47, NI-92/01

Integrable Generalisations of the 2-dimensional Born Infeld Equation

The Born-Infeld equation in two dimensions is generalised to higher dimensions whilst retaining Lorentz Invariance and complete integrability. This generalisation retains homogeneity in second derivatives of the field.Comment: 11 pages, Latex, DTP/93/3

Wigner Trajectory Characteristics in Phase Space and Field Theory

Exact characteristic trajectories are specified for the time-propagating Wigner phase-space distribution function. They are especially simple---indeed, classical---for the quantized simple harmonic oscillator, which serves as the underpinning of the field theoretic Wigner functional formulation introduced. Scalar field theory is thus reformulated in terms of distributions in field phase space. Applications to duality transformations in field theory are discussed.Comment: 9 pages, LaTex2

Deformation Quantization: Quantum Mechanics Lives and Works in Phase-Space

Wigner's quasi-probability distribution function in phase-space is a special (Weyl) representation of the density matrix. It has been useful in describing quantum transport in quantum optics; nuclear physics; decoherence (eg, quantum computing); quantum chaos; "Welcher Weg" discussions; semiclassical limits. It is also of importance in signal processing. Nevertheless, a remarkable aspect of its internal logic, pioneered by Moyal, has only emerged in the last quarter-century: It furnishes a third, alternative, formulation of Quantum Mechanics, independent of the conventional Hilbert Space, or Path Integral formulations. In this logically complete and self-standing formulation, one need not choose sides--coordinate or momentum space. It works in full phase-space, accommodating the uncertainty principle. This is an introductory overview of the formulation with simple illustrations.Comment: LaTeX, 22 pages, 2 figure

Self-dual Yang-Mills fields in pseudoeuclidean spaces

The self-duality Yang-Mills equations in pseudoeuclidean spaces of dimensions $d\leq 8$ are investigated. New classes of solutions of the equations are found. Extended solutions to the D=10, N=1 supergravity and super Yang-Mills equations are constructed from these solutions.Comment: 9 pages, LaTeX, no figure

Finite Euler Hierarchies And Integrable Universal Equations

Recent work on Euler hierarchies of field theory Lagrangians iteratively constructed {}from their successive equations of motion is briefly reviewed. On the one hand, a certain triality structure is described, relating arbitrary field theories, {\it classical\ts} topological field theories -- whose classical solutions span topological classes of manifolds -- and reparametrisation invariant theories -- generalising ordinary string and membrane theories. On the other hand, {\it finite} Euler hierarchies are constructed for all three classes of theories. These hierarchies terminate with {\it universal\ts} equations of motion, probably defining new integrable systems as they admit an infinity of Lagrangians. Speculations as to the possible relevance of these theories to quantum gravity are also suggested.Comment: (replaces previous unprintable version corrupted mailer) 13 p., (Plain TeX), DTP-92/3

Impact of multiscale dynamical processes and mixing on the chemical composition of the upper troposphere and lower stratosphere during the Intercontinental Chemical Transport Experiment–North America

We use high-frequency in situ observations made from the DC8 to examine fine-scale tracer structure and correlations observed in the upper troposphere and lower stratosphere during INTEX-NA. Two flights of the NASA DC-8 are compared and contrasted. Chemical data from the DC-8 flight on 18 July show evidence for interleaving and mixing of polluted and stratospheric air masses in the vicinity of the subtropical jet in the upper troposphere, while on 2 August the DC-8 flew through a polluted upper troposphere and a lowermost stratosphere that showed evidence of an intrusion of polluted air. We compare data from both flights with RAQMS 3-D global meteorological and chemical model fields to establish dynamical context and to diagnose processes regulating the degree of mixing on each day. We also use trajectory mapping of the model fields to show that filamentary structure due to upstream strain deformation contributes to tracer variability observed in the upper troposphere. An Eulerian measure of strain versus rotation in the large-scale flow is found useful in predicting filamentary structure in the vicinity of the jet. Higher-frequency (6–24 km) tracer variability is attributed to buoyancy wave oscillations in the vicinity of the jet, whose turbulent dissipation leads to efficient mixing across tracer gradients