Papers by Peter Vassiliou
Lecture Notes in Control and Information Sciences, 2010
Cartan's method of moving frames is briefly recalled in the context of immersed curves in the hom... more Cartan's method of moving frames is briefly recalled in the context of immersed curves in the homogeneous space of a Lie group G. The contact geometry of curves in low dimensional equi-affine geometry is then made explicit. This delivers the complete set of invariant data which solves the G-equivalence problem via a straightforward procedure, and which is, in some sense a supplement to the equivariant method of Fels and Olver. Next, the contact geometry of curves in general Riemannian manifolds (M; g) is described. For the special case in which the isometries of (M; g) act transitively, it is shown that the contact geometry provides an explicit algorithmic construction of the differential invariants for curves in M . The inputs required for the construction consist only of the metric g and a parametrisation of structure group S O(n); the group action is not required and no integration is involved. To illustrate the algorithm we explicitly construct complete sets of differential invariants for curves in the Poincar e half-space H 3 and in a family of constant curvature 3-metrics. It is conjectured that similar results are possible in other Cartan geometries. © : he grtn prolongtion we shll onsider is otined from thisX c F X¨v I C v S C v W C I @v U C Qv II A C P v IH ; @ 1 ; @ 2 © defined over G ¢ R P F o pply the generlised qourst norml formD heorem QD we work with this dul undle Xa c F nd lulte IT FtF ssiliou i @iA ghr @iA H¨v I C v S C v W C I @v U C Qv II A C P v IH ; @ 1 ; @ 2 © fHg I ¨¨v U C Qv II ; v IH © f@ 1 ; @ 2 g P @IA¨¨v R C Pv V ; v U v II © ghr @IA¨f v U C Qv II ; v IH g Q @PA¨¨v R Pv V ; v I C v S Sv W © ghr @PA¨f v R C Pv V ; v U v II g R @QA¨¨v T ; v I Qv S © ghr @QA¨f v R Pv V ; v W g S T @G ¢ R P A T @G ¢ R P A o hypotheses AD A nd A of heorem Q re stisfied with q a P nd derived length k a SF ine q > ID it remins to hek the singulr vriety of the quotient @RA =ghr @RA F prom the tle we see tht ghr @RA a¨v R ; v U ; v V ; v W ; v IH ; v II ; @ 1 ; @ 2 © nd @RA a¨v I ; v R ; v S ; v T ; v U ; v V ; v W ; v IH ; v II ; @ 1 ; @ 2 © nd hene @RA Xa @RA ghr @RA a¨v I ; v S ; v T © :
Symmetry Integrability and Geometry Methods and Applications, Oct 19, 2009
Cartan's method of moving frames is briefly recalled in the context of immersed curves in the hom... more Cartan's method of moving frames is briefly recalled in the context of immersed curves in the homogeneous space of a Lie group G. The contact geometry of curves in low dimensional equi-affine geometry is then made explicit. This delivers the complete set of invariant data which solves the G-equivalence problem via a straightforward procedure, and which is, in some sense a supplement to the equivariant method of Fels and Olver. Next, the contact geometry of curves in general Riemannian manifolds (M,g) is described. For the special case in which the isometries of (M,g) act transitively, it is shown that the contact geometry provides an explicit algorithmic construction of the differential invariants for curves in M. The inputs required for the construction consist only of the metric g and a parametrisation of structure group SO(n); the group action is not required and no integration is involved. To illustrate the algorithm we explicitly construct complete sets of differential invariants for curves in the Poincaré half-space H3 and in a family of constant curvature 3-metrics. It is conjectured that similar results are possible in other Cartan geometries.
Transactions of the American Mathematical Society, 2001
It is well known that if a scalar second order hyperbolic partial differential equation in two in... 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 common with infinite dimensional completely integrable systems.
Ijcm, Jun 11, 2004
Let $\CV$ be a vector field distribution on manifold $M$. We give an efficient algorithm for the ... more Let $\CV$ be a vector field distribution on manifold $M$. We give an efficient algorithm for the construction of local coordinates on $M$ such that $\CV$ may be locally expressed as some partial prolongation of the contact distribution $\Cal C^{(1)}_q$, on the first order jet bundle of maps from $\Bbb R$ to $\Bbb R^q$, $q\geq 1$. It is proven that if $\CV$ is locally equivalent to a partial prolongation of $\Cal C^{(1)}_q$ then the explicit construction of contact coordinates algorithmically depends upon the determination of certain first integrals in a sequence of geometrically defined and algorithmically determined integrable Pfaffian systems on $M$. The number of these first integrals that must be computed satisfies a natural minimality criterion. These results therefore provide a full and constructive generalisation of the classical Goursat normal form from the theory of exterior differential systems.
It is well known that if a scalar second order hyperbolic partial differential equation in two in... 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 common with infinite dimensional completely integrable systems.
Symmetry, Integrability and Geometry: Methods and Applications, 2009
Cartan's method of moving frames is briefly recalled in the context of immersed curves in the hom... more Cartan's method of moving frames is briefly recalled in the context of immersed curves in the homogeneous space of a Lie group G. The contact geometry of curves in low dimensional equi-affine geometry is then made explicit. This delivers the complete set of invariant data which solves the G-equivalence problem via a straightforward procedure, and which is, in some sense a supplement to the equivariant method of Fels and Olver. Next, the contact geometry of curves in general Riemannian manifolds (M, g) is described. For the special case in which the isometries of (M, g) act transitively, it is shown that the contact geometry provides an explicit algorithmic construction of the differential invariants for curves in M . The inputs required for the construction consist only of the metric g and a parametrisation of structure group SO(n); the group action is not required and no integration is involved. To illustrate the algorithm we explicitly construct complete sets of differential invariants for curves in the Poincaré half-space H 3 and in a family of constant curvature 3-metrics. It is conjectured that similar results are possible in other Cartan geometries.
Symmetry, Integrability and Geometry: Methods and Applications, 2013
The Cauchy problem for harmonic maps from Minkowski space with its standard flat metric to a cert... 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 differential equation of Lie type associated to SL(2) acting on a manifold of dimension 4. This is further reduced to the simplest Lie system: the Riccati equation. Lie reduction permits explicit representation formulas for various initial value problems. Additionally, a concise (hyperbolic) Weierstrass-type representation formula is derived. Finally, a number of open problems are framed.
Studies in Applied Mathematics, 2011
Journal of Geometry and Physics, 2008
Foundations of Computational Mathematics, 2006
Let V be a vector field distribution or Pfaffian system on manifold M. We give an efficient algor... more Let V be a vector field distribution or Pfaffian system 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 q , on the first-order jet bundle of maps from R to R q , q ≥ 1. It is proven that if V is locally equivalent to a partial prolongation of C (1) q , then the explicit construction of contact coordinates algorithmically depends upon the determination of certain first integrals in a sequence of geometrically defined and algorithmically determined integrable Pfaffian systems on M. The number of these first integrals that must be computed satisfies a natural minimality criterion. These results provide a full and constructive generalisation of the Goursat normal form from the theory of exterior differential systems.
Differential Geometry and its Applications, 2006
We provide necessary and sufficient conditions on the derived type of a vector field distribution... more 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
Bulletin of the Australian Mathematical Society, 1990
Bulletin of the Australian Mathematical Society, 1994
We define the notion of Darboux integrability for linear second order partial differential operat... more We define the notion of Darboux integrability for linear second order partial differential operators,
Bulletin of the Australian Mathematical Society, 1987
Service Announcement Due to essential infrastructure maintenance work, users may experience inter... more Service Announcement Due to essential infrastructure maintenance work, users may experience intermittent access issues on Tuesday14th June 2011 between 7: 00am GMT and 8: 00am GMT. We apologise for any inconvenience this may cause.
The ANZIAM Journal, 2002
We give an intrinsic construction of a coupled nonlinear system consisting of two firstorder part... more We give an intrinsic construction of a coupled nonlinear system consisting of two firstorder partial differential equations in two dependent and two independent variables which is determined by a hyperbolic structure on the complex special linear group regarded as a real Lie group G. Despite the fact that the system is not Darboux semi-integrable at first order, the construction of a family of solutions depending upon two arbitrary functions, each of one variable, is reduced to a system of ordinary differential equations on the 1-jets. The ordinary differential equations in question are of Lie type and associated with G. Serial-fee code 1446-8735/02 3 However, there is currently a vigorous research program whose aim is to find new reductions beyond those provided by classical Lie methods. Apart from [8], see, for example, the non-exhaustive list [1, 6, 7], and references therein. 83 84 Peter J. Vassiliou can be shown ([5, 9]) that the general solution of the system can be constructed from an ODE system and this ODE system may itself be canonically constructed without having to solve PDE systems. Thus if a hyperbolic first-order system is Darboux semiintegrable, then the construction of its general solution, that is, a family of smooth solutions parametrised by two arbitrary smooth functions each of one variable, is entirely reduced to ODEs. Roughly speaking, a first-order system is Darboux semi-integrable (at first order) if one of its characteristic systems has at least two independent characteristic invariants 4 (first-order differential functions which are constant along the system's characteristic flows). An interesting question is the converse: if the construction of the general solution of an (intrinsically) nonlinear hyperbolic system is reducible to ODEs then is the system Darboux semi-integrable? In this paper we establish the existence of a nonlinear, nondegenerate 5 first-order hyperbolic system, the construction of whose general solution is reduced to an ODE, and yet the system has no regular first-order invariants on either characteristic system. That is, a nonlinear first-order system which is not Darboux semi-integrable at first order but whose general solution can nevertheless be constructed by solving an ODE on the 1-jets. 6 Another interesting feature of our example arises from the following consideration: it can, in fact, be shown that the constructed system is Darboux semi-integrable at higher order, namely, at order 3. Our construction thus foreshadows the system's integrability at lowest possible order. This begins to address one of the basic open problems in geometric integrability, namely, to find a low order test for Darboux semi-integrability.
Applicable Algebra in Engineering, Communication and Computing, 2001
ABSTRACT . Hyperbolic systems of first order partial differential equations in two dependent and ... more ABSTRACT . 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 computing the tangential characteristic symmetries of a certain Fermi–Pasta–Ulam equation. As a consequence we reduce its associated intrinsic hyperbolic structure to Vessiot normal form and prove that this equation may be regarded as an ordinary differential equation on ℜ2. This result has a far reaching generalisation in which the well known local linearisation of this equation by a hodograph transformation is shown to be a special case of a theorem due to Vessiot. This is further illustrated by studying the Born-Infeld system of Arik, Neyzi, Nutku, Olver and Verosky and the s = 0 Liouville system of Bryant, Griffiths and Hsu. It is shown that the latter may be regarded as an ordinary differential equation on the Lie group of affine transformations of the real line.
... au Mark E. Johnston Department of Applied Mathematics and Theoretical Physics, University of ... more ... au Mark E. Johnston Department of Applied Mathematics and Theoretical Physics, University of Cambridge ... au Peter J. Vassiliou School of Mathematics and Statistics, University of Canberra, Australian ... elementary introduction to some of the basic ideas in the subject as a whole. ...
Acta Applicandae Mathematicae, 1987
We study coupled systems of nonlinear wave equations from the point of view of their formal Darbo... more We study coupled systems of nonlinear wave equations from the point of view of their formal Darboux integrability. By making use of Vessiot's geometric theory of differential equations, it is possible to associate to each system of nonlinear wave equations a module of vector fields on the second-order jet bundle -the Vessiot distribution. By imposing certain conditions of the structure of the Vessiot distributions, we identify the so-called separable Vessiot distributions. By expressing the separable Vessiot distributions in a basis of singular vector fields, we show that there are, at most, 27 equivalence classes of such distributions. Of these, 14 classes are associated with Darboux integrable nonlinear systems. We take one of these Darboux integrable classes and show that it is in correspondence with the class of six-dimensional simply transitive Lie algebras. Finally, this later result is used to reduce the problem of constructing exact general solutions of the nonlinear wave equations understudy to the integration of Lie systems. These systems were first discovered by Sophus Lie as the most general class of ordinary differential equations which admit nonlinear superposition principles.
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Papers by Peter Vassiliou