# Peter J VassiliouAustralian National University | ANU · The Mathematical Sciences Institute

Peter J Vassiliou

PhD

## About

44

Publications

3,838

Reads

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350

Citations

Citations since 2016

Introduction

I am interested in differential geometry and its application to differential equations, integrable systems (especially Darboux integrable systems), control theory and classical physics. I am happiest when interesting geometric structures can be used to gain insight into physical or geometric phenomena.

**Skills and Expertise**

## Publications

Publications (44)

The purpose of this note is to describe a recent generalisation of the well-known Goursat normal form and explore its possible
role in control theory. For instance, we give a new, straightforward, general procedure for linearising nonlinear control
systems, including time-varying, fully nonlinear systems and we illustrate the method by elementary p...

Let Pfaffian system ω define an intrinsically nonlinear control system which is invariant under a Lie group of symmetries G. Using the contact geometry of Brunovsky normal forms and symmetry reduction, this paper solves the problem of constructing subsystems α ⊂ ω such that α defines a static feedback linearizable control system. A method for repre...

Given a smooth 2-dimensional Riemannian or pseudo-Riemannian manifold $(M, \boldsymbol{g})$ and an ambient 3-dimensional Riemannian or pseudo-Riemannian manifold $(N, \boldsymbol{h})$, one can ask under what circumstances does the exterior differential system $\mathcal{I}$ for the isometric embedding $M\hookrightarrow N$ have particularly nice solv...

Control systems of interest are often invariant under Lie groups of transformations. Given such a control system, assumed to not be static feedback linearizable, a verifiable geometric condition is described and proven to guarantee its dynamic feedback linearizability. Additionally, a systematic procedure for obtaining all the system trajectories i...

The class of separable solutions of a 1-dimensional sourceless diffusion equation is stabilized by the action of the generic symmetry group. It includes all solutions invariant under a subgroup of the generic group. An equation which admits separation of variables in some field coordinate has separable solutions not invariant under any subgroup, as...

Let Pfaffian system ω define an intrinsically nonlinear control system on manifold M that is invariant under the free, regular action of a Lie group G. The problem of identifying and constructing static feedback linearizable G-quotients of ω was solved in De Doná et al. (2016). Building on these results, the present paper proves that the trajectori...

Let Pfaffian system ${\omega}$ define an intrinsically nonlinear control system which is invariant under a Lie group of symmetries $G$. Using the contact geometry of Brunovsky normal forms and symmetry reduction, this paper solves the problem of constructing subsystems ${\alpha}\subset{\omega}$ such that ${\alpha}$ defines a static feedback lineari...

We study control systems invariant under a Lie group with application to the problem of nonlinear trajectory planning. A theory of symmetry reduction of exterior differential systems [2] is employed to demonstrate how symmetry reduction and reconstruction is effective in the explicit, exact construction of planned system trajectories. We show that,...

The recognition problem for differential systems locally equivalent to the contact system for curves is solved in terms of numerical invariants - the derived type. This gives a complete generalization of the classical Goursat normal form. The proof is constructive and has applications in control theory and in the solution of Darboux integrable syst...

It is shown that there are nonlinear sigma models which are Darboux
integrable and possess a solvable Vessiot group in addition to those whose
Vessiot groups are central extensions of semi-simple Lie groups. They govern
harmonic maps between Minkowski space $\mathbb{R}^{1,1}$ and certain complete,
non-constant curvature 2-metrics. The solvability o...

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...

The wave equation $u_{tt} = c^2 u_{xx}$ is generally regarded as a linear
approximation to the equation describing the amplitude of a transversely
vibrating elastic string in the plane. But, as is shown in \cite{BC96}, the
assumption of transverse vibration in fact implies that the wave equation
describes the vibration precisely, with no need for a...

With few exceptions, known explicit solutions of the curve shortening flow
(CSE) of a plane curve, can be constructed by classical Lie point symmetry
reductions or by functional separation of variables. One of the functionally
separated solutions is the exact curve shortening flow of a closed, convex
"oval"-shaped curve and another is the smoothing...

The intrinsic geometric properties of generalized Darboux-Manakov-Zakharov systems of semilinear partial differential equations
for a real-valued function u(x1, …, xn) are studied with particular reference to the linear systems in this equation class.
System (1) is overdetermined and will not generally be involutive in the sense of Cartan: its coe...

The intrinsic geometric properties of generalized Darboux-Manakov-Zakharov systems of semilinear partial differential equations \label{GDMZabstract} \frac{\partial^2 u}{\partial x_i\partial x_j}=f_{ij}\Big(x_k,u,\frac{\partial u}{\partial x_l}\Big), 1\leq i<j\leq n, k,l\in\{1,...,n\} for a real-valued function $u(x_1,...,x_n)$ are studied with part...

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, a...

In this article we solve an inverse problem in the theory of quotients for differential equations. We characterize a family of exterior differential systems that can be written as a quotient of a direct sum of two associated systems that are constructed from the original. The fact that a system can be written as a quotient can be used to find the g...

The notions of weak Darboux integrability and hyperbolic reduction are introduced, and their potential is gauged as a means of extending the range of application of geometric methods for solving hyperbolic partial differential equations. For directness, our work is expressed in local coordinates and formulated for semilinear hyperbolic systems in t...

In this paper we present a far-reaching generalization of E. Vessiot's analysis 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 Darboux integrable systems, and provides for systemat...

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(1)q, on the first-order jet bundle of maps from ℝ to ℝq, q ≥ 1. It is proven that if V is locally equival...

We provide necessary and sufficient conditions on the derived type of a vector field distribution $\Cal V$ in order that it be locally equivalent to a partial prolongation of the contact distribution $\Cal C^{(1)}_q$, on the first order jet bundle of maps from $\Bbb R$ to $\Bbb R^q$, $q\geq 1$. This result fully generalises the classical Goursat no...

We give an intrinsic construction of a coupled nonlinear system consisting of two first-order 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...

We give an intrinsic construction of a coupled nonlinear system consisting of two first-order 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...

. 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...

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

Linearizable discrete dynamical systems (DDS) provide a valuable class of examples for any a priori test of integrability. In this paper, we give an overview of existence results for Lie symmetries and how they lead to (local, possibly partial) linearizations of DDS. We also give a classification of DDS that are globally linearized by single-valued...

. The singularity manifold equation of the Kadomtsev-Petviashvili equation, the socalled Krichever-Novikov equation, has an exact linearization to an overdetermined system of partial differential 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 differen...

If a first-order discrete dynamical system (DDS) possesses a continuous symmetry, the system is known to be linearizable. In this paper, we show that a first-order analytic DDS always possesses a symmetry in a neighbourhood of a hyperbolic fixed point.

We define the notion of Darboux integrability for linear second order partial differential operators,
.
We then build on certain geometric results of E. Vessiot related to the theory of Monge characteristics to show that the Darboux integrable operators L can be used to obtain a solution of the A 2 Toda field theory. This solution is parametrised b...

Invariance of the (2 + l)-dimensional Harry-Dym equation under a novel reciprocal transformation is shown to encode a linear representation and to generate auto-Bäcklund transformations for the (2 + l)-dimensional Krichever-Novikov, mKP and KP equations. A linear decomposition of the (2 + 1)-dimensional Krichever-Novikov equation is used to solve c...

The oldest technique available for constructing the explicit solution of the Cauchy problem for systems of hyperbolic partial differential equations is the method of characteristics [1]. In his classic study [2] of 1860, Riemann wrote the equations of 1-dimensional compressible fluid flow in so called Riemann invariants. He was able to then lineari...

The main aim of this paper is to study the Cauchy problem for Riemann double waves. That is, given a hyperbolic system of quasilinear partial differential equations, we derive sufficient conditions in the Cauchy problem in order that the solution will time-evolve as a Riemann double wave. These conditions in general can only be given in implicit fo...

We give sufficient conditions for C∞ vector field systems on Rn with genus g = 1 to be diffeomorphic to a contact structure. The diffeomorphism is explicitly constructed and used to give the most general integral submanifolds for the systems. Finally the implications of these results for integrable hyperbolic partial differential equations in the p...

The geometry of nonlinear wave equations in two independent variables: methods for exact solution - Volume 36 Issue 3 - Peter John Vassiliou

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 impos...

In the first paper of this series a correspondence was established between coupled systems of two-dimensional nonlinear wave equations and the six-dimensional simply transitive Lie algebras. In the present paper we make use of this result to construct a Darboux integrable and exactly integrable nonlinear system associated with the six-parameter nil...

Weintroduce the notion of generalized Darboux-Manakov-Zakharov sys- tems of semilinear partial differential equations. ∂2u ∂ xi∂ x j = f ij

## Projects

Projects (2)

For control systems with a Lie group of symmetries we are able to decompose trajectory generation to that of the composition of a linearizable control subsystem and a control system on its Lie group of symmetries. The aim of the project is to use this to represent trajectories using the minimum possible quadrature and/or to explicitly construct the maximal quadrature fee trajectories.