Peter J. Vassiliou

Separation of variables for the 1-dimensional non-linear diffusion equation

Abstract 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 in the linear case. The class of separable equations significantly extends the class of equations having non-generic symmetry, i.e. those with exponential or power law diffusivities, for which separation of variables is a trivial process. We derive a complete list of canonical forms for diffusion equations which admit separation of variables in some coordinate, and we describe the separation mechanism for these equations. It involves the integration of a fixed third order ordinary differential equation, generally non-linear, and the subsequent integration of a first order ordinary differential equation which depends on the particular solution of the third order equation. The procedure yields a 3-parameter family of separable solutions of the given diffusion equation. Several non-symmetric examples are analyzed in detail, leading to explicit non-invariant solutions.

Vessiot structure for manifolds of (,)-hyperbolic type: Darboux integrability and symmetry

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. In this paper we employ a geometric object intrinsically associated with any hyperbolic partial differential equation, its hyperbolic structure, to study the Darboux integrability of the class E of semilinear second order hyperbolic partial differential equations in one dependent and two independent variables. It is shown that the problem of classifying the Darboux integrable equations in E contains, as a subproblem, that of classifying the manifolds of (p, q)hyperbolic type of rank 4 and dimension 2k + 3, k ≥ 2; p = 2, q ≥ 2. In turn, it is shown that the problem of classifying these manifolds in the two (lowest) cases (p, q) = (2, 2), (2, 3) contains, as a subproblem, the classification problem for Lie groups. This generalizes classical results of E. Vessiot. The main result is that if an equation in E is (2,2)or (2,3)-Darboux integrable on the k-jets, k ≥ 2, then its intrinsic hyperbolic structure admits a Lie group of symmetries of dimension 2k− 1 or 2k− 2, respectively. It follows that part of the moduli space for the Darboux integrable equations in E is determined by isomorphism classes of Lie groups. The Lie group in question is the group of automorphisms of the characteristic systems of the given equation which leaves invariant the foliation induced by the characteristic (or, Riemann) invariants of the equation, the tangential characteristic symmetries. The isomorphism class of the tangential characteristic symmetries is a contact invariant of the corresponding Darboux integrable partial differential equation.

Efficient Construction of Contact Coordinates for Partial Prolongations

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 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.

Tangential Characteristic Symmetries and First Order Hyperbolic Systems

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.

An invariance of the (2 + 1)-dimensional Harry-Dym equation: Application to initial/boundary value problems

Abstract 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-Backlund 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 classes of initial/boundary value problems both on the half-plane and on the quarter-plane.

Cauchy Problem for a Darboux Integrable Wave Map System and Equations of Lie Type

The Cauchy problem for harmonic maps from Minkowski space with its stan- dard 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 ener- gy 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. Addi- tionally, a concise (hyperbolic) Weierstrass-type representation formula is derived. Finally, a number of open problems are framed.

Contact Geometry and Its Application to Control

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 pedagogical examples. We also exhibit an apparently non-flat control system which can nevertheless be explicitly linearised and therefore posseses an infinite symmetry group.

Contact Geometry of Curves

Cartans 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 Poincare 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 Reduction, Contact Geometry, and Partial Feedback Linearization

Let Pfaffian system

Formal analysis of the Cauchy problem for a system associated with the (2+1)-dimensional Krichever-Novikov equation

{\omega}