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We provide necessary and sufficient conditions on the derived type of a vector field distribution V in order that it be locally equivalent to a partial prolongation of the contact distribution C (1) q , on the 1 order jet bundle of maps from R to R, q ≥ 1. This result fully generalises the classical Goursat normal form. Our proof is constructive: it is… (More)

- Peter J. Vassiliou
- Foundations of Computational Mathematics
- 2006

Let V be a vector field distribution on manifold M . We give an efficient algorithm for the construction of local coordinates on M such that V may be locally expressed as some partial prolongation of the contact distribution C (1) q , on the 1 st order jet bundle of maps from R to R, q ≥ 1. It is proven that if V is locally equivalent to a partial… (More)

It is well known that if a scalar second order hyperbolic partial differential equation in two independent variables is Darboux integrable, then its local Cauchy problem may be solved by ordinary differential equations. In addition, such an equation has infinitely many non-trivial conservation laws. Moreover, Darboux integrable equations have properties in… (More)

We generalise the notion of contact manifold by allowing the contact distribution to have codimension two. There are special features in dimension six. In particular, we show that the complex structure on a three-dimensional complex contact manifold is determined solely by the underlying contact distribution.

- Peter J. Vassiliou
- Applicable Algebra in Engineering, Communication…
- 2001

Hyperbolic systems of first order partial differential equations in two dependent and two independent variables are studied from the point of view of their local geometry. We illustrate an earlier result on such systems, which derived a complete set of local invariants for the class of systems which are (2,2)-Darboux integrable on the 1-jets, by explicitly… (More)

We generalise the notion of contact manifold by allowing the contact distribution to have codimension two. There are special features in dimension six. In particular, we show that the complex structure on a three-dimensional complex contact manifold is determined solely by the underlying contact distribution. Definitions Suppose M is a 6-dimensional… (More)

The singularity manifold equation of the Kadomtsev-Petviashvili equation, the so-called Krichever-Novikov equation, has an exact linearization to an overdetermined system of partial diierential equations in three independent variables. We study in detail the Cauchy problem for this system as an example for the use of the formal theory of diierential… (More)

The Cauchy problem for harmonic maps from Minkowski space with its standard flat metric to a certain non-constant curvature Lorentzian 2-metric is studied. The target manifold is distinguished by the fact that the Euler–Lagrange equation for the energy functional is Darboux integrable. The time evolution of the Cauchy data is reduced to an ordinary… (More)

1 Introduction 1 1 Introduction In this paper we present a far-reaching generalization of E. Vessiot's analysis [24], [25] of the Darboux integrable partial differential equations in one dependent and two independent variables. Our approach provides new insights into this classical method, uncovers the fundamental geometric invariants of Dar-boux integrable… (More)