## 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.