Summary of research program

Prof. Raymond G. McLenaghan

The main theme of my research is the study of properties of differential equations arising in mathematical physics using the methods of differential geometry and Lie group theory. The work is being conducted in three inter-related areas the common threads of which are the geometrical and group invariant methods used.

The first area is that of completely integrable Hamiltonian systems. Such systems include a large number of important physical models described in general by nonlinear systems of differential equations that can be integrated by quadratures. To solve the problem of integrability for such systems defined on pseudo-Riemannian manifolds of low dimensions I have been using the method of moving frames and an analogue of the classical theory of algebraic invariants, based on group invariants of Killing tensors defined in pseudo-Riemannian spaces of constant curvature [2,4-6,8,12,15,17,18,20-22,24]. Related to this work is the problem of characterizing the R-separable coordinate systems for the Laplace equation by invariants of conformal Killing tensors [3].

The second area, which is closely related to the first, concerns separability theory for the Dirac equation on pseudo-Riemannian background spaces. Compared with the well developed theory of separation of variables which exists for the Hamilton-Jacobi (HJ) and Schrodinger (S) equations the parallel theory for the Dirac equation is an early stage of development. Starting from results due to Miller, I have been developing a link between Dirac separability and that of the HJ and S equations. A study of the Dirac equation on two-dimensional background spaces is being undertaken to give insight into the structure of the theory [18,24].

The third area of study concerns Hadamard's problem of diffusion of waves which consists of the determination of all the second order linear partial differential equations of normal hyperbolic type that satisfy Huygens' principle in the strict sense [1,28]. The physical significance of the Huygens' property is that the wave phenomena governed by the equation propagate sharply without a tail. Work is progressing on a proof of a generalized Hadamard conjecture for the self-adjoint equation for the physically interesting case of four dimensions. The conjecture states that a Huygens' equation is equivalent to the pure equation on an exact plane wave space-time. For the case of the non-self-adjoint equation I am studying the significance of a new non-trivial Huygens' equation which has recently been brought to light [1].

Note: The reference numbers refer to the articles listed in Recent publications (PDF).

My research is funded by an NSERC Discovery Grant.