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D. B. Fairlie

Publications and source records attributed to D. B. Fairlie.

29 records · Page 2Linked to original sources
arXiv1995

The Hamiltonian structure of the dispersionless Toda hierarchy

The Hamiltonian structure of the two-dimensional dispersionless Toda hierarchy is studied, this being a particular example of a system of hydrodynamic type. The polynomial conservation laws for the system turn out, after a change of variable, to be associated with the axially symmetric solutions of the 3-dimensional Laplace equation and this enables a generating function for the Hamiltonian densities to be derived in closed form.

solv-int↗
arXiv1994

A Fresh Look at Generalized Veneziano Amplitudes

The dual resonance model, which was a precursor of string theory was based upon the idea that two-particle scattering amplitudes should be expressible equivalently as a sum of contributions of an infinite number of $s$ channel poles each corresponding to a finite number of particles with definite spin, or as a similar sum of $t$ channel poles. The famous example of Veneziano \cite{ven} satisfies all these requirements, and is additionally ghost free.We recall other trajectories which provide solutions to the duality constraints, e.g. the general Mobiüs trajectories and the logarithmic trajectories, which were thought to be lacking this last feature. We however demonstrate, partly empirically, the existence of a regime within a particular deformation of the Veneziano amplitude for logarithmic trajectories for which the amplitude remains ghost free.

hep-th↗
arXiv1994

On Certain Quantum Deformations of $gl(N,R)$

In this paper all deformations of the general linear group, subject to certain restrictions which in particular ensure a smooth passage to the Lie group limit, are obtained. Representations are given in terms of certains sets of creation and annihilation operators. These creation and annihilation operators may belong to a generalisation of the $q$-quark type or $q$-hadronic type, of $q$-boson or $q$-fermion type. We are also led to a natural definition of $q$-direct sums of q-algebras.

hep-th↗
arXiv1994

General solutions of the Monge-Ampère equation in $n$-dimensional space

It is shown that the general solution of a homogeneous Monge-Ampère equation in $n$-dimensional space is closely connected with the exactly (but only implicitly) integrable system \frac {\partial ξ_{j}}{\partial x_0}+\sum_{k=1}^{n-1} ξ_{k} \frac {\partial ξ_{j}}{\partial x_{k}}=0 \label{1} Using the explicit form of solution of this system it is possible to construct the general solution of the Monge-Ampère equation.

hep-th↗
arXiv1993

Equations of Hydrodynamic Type

The universal field equations introduced by the author and his collaborators, which admit infinitely many inequivalent Lagrangian formulations are shown to arise as consistency conditions for the existence of non-trivial solutions to the quasi-linear equations, called equations of hydrodynamic type by Novikov , Dubrovin and others. The solutions in closed form are only implicit. A method due to Stokes, which is in essence just Fourier Analysis is resurrected for application to those equations. With the benefit of algebraic computation facilities, this method, allows the general structure of power series solutions to be conjectured.

hep-th↗
arXiv1992

Euler Hierarchies and Universal Equations

Finite Euler hierarchies of field theory Lagrangians leading to universal equations of motion for new types of string and membrane theories and for {\it classical} topological field theories are constructed. The analysis uses two main ingredients. On the one hand, there exists a generic finite Euler hierarchy for one field leading to a universal equation which generalises the Plebanski equation of self-dual four dimensional gravity. On the other hand, specific maps are introduced between field theories which provide a ``triangular duality'' between certain classes of arbitrary field theories, classical topological field theories and generalised string and membrane theories. The universal equations, which derive from an infinity of inequivalent Lagrangians, are generalisations of certain reductions of the Plebanski and KdV equations, and could possibly define new integrable systems, thus in particular integrable membrane theories. Some classes of solutions are constructed in the general case. The general solution to some of the universal equations is given in the simplest cases.

hep-th↗
arXiv1992

Universal Field Equations with Reparametrisation Invariance

New reparametrisation invariant field equations are constructed which describe $d$-brane models in a space of $d+1$ dimensions. These equations, like the recently discovered scalar field equations in $d+1$ dimensions, are universal, in the sense that they can be derived from an infinity of inequivalent Lagrangians, but are nonetheless Lorentz (Euclidean) invariant. Moreover, they admit a hierarchical structure, in which they can be derived by a sequence of iterations from an arbitrary reparametrisation covariant Lagrangian, homogeneous of weight one. None of the equations of motion which appear in the hierarchy of iterations have derivatives of the fields higher than the second. The new sequence of Universal equations is related to the previous one by an inverse function transformation. The particular case of $d=2$, giving a new reparametrisation invariant string equation in 3 dimensions is solved.

hep-th↗