My general field of research is in dynamical systems, and the
following is a brief overview of some major themes; the main emphasis
is on analytical and numerical studies of partial differential
equations exhibiting spatiotemporally complex and chaotic dynamics,
but applications to other fields have also captured my attention...
I welcome inquiries from potential students interested in these or
Pattern Formation and Spatiotemporal Chaos:
Numerous partial differential equations (PDEs) arising in contexts
such as fluid dynamics or surface growth display surprisingly complex
temporal dynamics and/or spatial pattern formation. The Kuramoto-Sivashinsky
is a particularly rich (and much-studied) example,
and I have long been interested in investigating various aspects of
spatiotemporal chaos in the KS equation and its generalizations,
analytically and numerically.
In recent years I have focussed especially on a related 6th-order PDE,
the Nikolaevskiy model for short-wave pattern formation with Galilean
invariance, and its associated Matthews-Cox modulation
equations. Work with my former student Philip Poon revealed
spatiotemporal chaos with strong scale separation, potential anomalous
scaling, Burgers-like viscous shocks and coarsening phenomena to
chaos-stabilized fronts; there is much that remains to be done to
understand this curious dynamical behaviour!
While the solutions of such nonlinear PDEs are typically too complex
to permit detailed analytical description, rigorous
functional-analytic estimates on global, long-time or averaged
properties of solutions on the attractor may nevertheless often be
am especially interested in the interplay between numerical and
analytical results; as an example, my numerical discovery and
asymptotic investigation of a viscous shock solution in the
destabilized KS equation influenced subsequent improvements in, and
constraints on, rigorous bounds on the scaling of the absorbing ball
for the KS equation.
A related major theme of my research concerns analytical estimates in
fluid dynamics, notably turbulent Rayleigh-Bénard convection, for
which I am particularly interested in establishing rigorous a
variational bounds on averaged quantities such as bulk
convective heat transport.
I have collaborated on and (co-)supervised students interested in
dynamical models in various areas, including mathematical epidemiology
and immunology, aggregation models and opinion dynamics. For
much of this research, I am associated with the IMPACT-HIV
group (based at the IRMACS Centre
at SFU), an interdisciplinary research team studying differential
equation and network models of the HIV epidemic, with a particular
focus on evaluating Treatment as Prevention control strategies.